Termination w.r.t. Q proof of /tmp/tmpu2J_tX/OvConsOS_complete

Termination w.r.t. Q of the following Term Rewriting System could not be shown:

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, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
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)))
isNatIListKind(n__take(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))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
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)))
isNatIListKind(n__take(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))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X

Q is empty.

Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

ISNATILIST(V) → ACTIVATE(V)
ISNATLIST(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ISNATILISTKIND(n__take(V1, V2)) → ACTIVATE(V1)
ISNATILISTKIND(n__take(V1, V2)) → ISNATKIND(activate(V1))
U62¹(tt, V2) → U63¹(isNatIList(activate(V2)))
ISNATILISTKIND(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ACTIVATE(n__0) → 0¹
ISNATKIND(n__length(V1)) → ACTIVATE(V1)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATILIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
LENGTH(cons(N, L)) → ISNAT(N)
ISNAT(n__length(V1)) → U11¹(isNatIListKind(activate(V1)), activate(V1))
ISNATKIND(n__length(V1)) → ISNATILISTKIND(activate(V1))
ACTIVATE(n__length(X)) → LENGTH(X)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
U11¹(tt, V1) → U12¹(isNatList(activate(V1)))
TAKE(0, IL) → ISNATILIST(IL)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
TAKE(s(M), cons(N, IL)) → AND(isNat(M), n__isNatKind(M))
U61¹(tt, V1, V2) → ACTIVATE(V2)
U42¹(tt, V2) → ACTIVATE(V2)
U52¹(tt, V2) → ACTIVATE(V2)
U41¹(tt, V1, V2) → ISNAT(activate(V1))
U21¹(tt, V1) → ISNAT(activate(V1))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
TAKE(s(M), cons(N, IL)) → AND(isNatIList(activate(IL)), n__isNatIListKind(activate(IL)))
TAKE(0, IL) → AND(isNatIList(IL), n__isNatIListKind(IL))
U51¹(tt, V1, V2) → ACTIVATE(V1)
U91¹(tt, IL, M, N) → CONS(activate(N), n__take(activate(M), activate(IL)))
TAKE(s(M), cons(N, IL)) → AND(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))))
ISNATILIST(n__cons(V1, V2)) → U41¹(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U91¹(tt, IL, M, N) → ACTIVATE(M)
LENGTH(cons(N, L)) → AND(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N)))
TAKE(s(M), cons(N, IL)) → ISNAT(N)
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V1)
U11¹(tt, V1) → ACTIVATE(V1)
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V2)
U41¹(tt, V1, V2) → ACTIVATE(V1)
U51¹(tt, V1, V2) → U52¹(isNat(activate(V1)), activate(V2))
U21¹(tt, V1) → ACTIVATE(V1)
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
U42¹(tt, V2) → U43¹(isNatIList(activate(V2)))
U71¹(tt, L) → ACTIVATE(L)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ZEROS → CONS(0, n__zeros)
U91¹(tt, IL, M, N) → ACTIVATE(IL)
U11¹(tt, V1) → ISNATLIST(activate(V1))
U51¹(tt, V1, V2) → ACTIVATE(V2)
U71¹(tt, L) → LENGTH(activate(L))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
U61¹(tt, V1, V2) → ISNAT(activate(V1))
U52¹(tt, V2) → ISNATLIST(activate(V2))
U62¹(tt, V2) → ISNATILIST(activate(V2))
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V2)
U91¹(tt, IL, M, N) → ACTIVATE(N)
U42¹(tt, V2) → ISNATILIST(activate(V2))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__cons(X1, X2)) → CONS(X1, X2)
ISNATILISTKIND(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATILISTKIND(n__take(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ISNATLIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
LENGTH(nil) → 0¹
ACTIVATE(n__zeros) → ZEROS
U81¹(tt) → NIL
U52¹(tt, V2) → U53¹(isNatList(activate(V2)))
LENGTH(cons(N, L)) → U71¹(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
ACTIVATE(n__nil) → NIL
U61¹(tt, V1, V2) → ACTIVATE(V1)
U61¹(tt, V1, V2) → U62¹(isNat(activate(V1)), activate(V2))
ACTIVATE(n__s(X)) → S(X)
ISNAT(n__s(V1)) → ACTIVATE(V1)
AND(tt, X) → ACTIVATE(X)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
ISNATILIST(V) → ISNATILISTKIND(activate(V))
ISNATLIST(n__take(V1, V2)) → ISNATKIND(activate(V1))
ISNAT(n__s(V1)) → U21¹(isNatKind(activate(V1)), activate(V1))
U31¹(tt, V) → ISNATLIST(activate(V))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ISNATLIST(n__take(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V1)
ISNATILISTKIND(n__take(V1, V2)) → ACTIVATE(V2)
U41¹(tt, V1, V2) → U42¹(isNat(activate(V1)), activate(V2))
U21¹(tt, V1) → U22¹(isNat(activate(V1)))
ISNAT(n__length(V1)) → ISNATILISTKIND(activate(V1))
U31¹(tt, V) → ACTIVATE(V)
U31¹(tt, V) → U32¹(isNatList(activate(V)))
TAKE(0, IL) → U81¹(and(isNatIList(IL), n__isNatIListKind(IL)))
TAKE(s(M), cons(N, IL)) → ISNAT(M)
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
U41¹(tt, V1, V2) → ACTIVATE(V2)
U51¹(tt, V1, V2) → ISNAT(activate(V1))
LENGTH(cons(N, L)) → AND(isNatList(activate(L)), n__isNatIListKind(activate(L)))
LENGTH(cons(N, L)) → ACTIVATE(L)
ISNATLIST(n__cons(V1, V2)) → U51¹(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
ISNATLIST(n__take(V1, V2)) → U61¹(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
TAKE(s(M), cons(N, IL)) → U91¹(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ISNATILIST(V) → U31¹(isNatIListKind(activate(V)), activate(V))
U71¹(tt, L) → S(length(activate(L)))
U62¹(tt, V2) → ACTIVATE(V2)
ACTIVATE(n__take(X1, X2)) → TAKE(X1, X2)
ISNATILIST(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ISNAT(n__length(V1)) → ACTIVATE(V1)
ZEROS → 0¹

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, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
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)))
isNatIListKind(n__take(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))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

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

ISNATILIST(V) → ACTIVATE(V)
ISNATLIST(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ISNATILISTKIND(n__take(V1, V2)) → ACTIVATE(V1)
ISNATILISTKIND(n__take(V1, V2)) → ISNATKIND(activate(V1))
U62¹(tt, V2) → U63¹(isNatIList(activate(V2)))
ISNATILISTKIND(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ACTIVATE(n__0) → 0¹
ISNATKIND(n__length(V1)) → ACTIVATE(V1)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATILIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
LENGTH(cons(N, L)) → ISNAT(N)
ISNAT(n__length(V1)) → U11¹(isNatIListKind(activate(V1)), activate(V1))
ISNATKIND(n__length(V1)) → ISNATILISTKIND(activate(V1))
ACTIVATE(n__length(X)) → LENGTH(X)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
U11¹(tt, V1) → U12¹(isNatList(activate(V1)))
TAKE(0, IL) → ISNATILIST(IL)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
TAKE(s(M), cons(N, IL)) → AND(isNat(M), n__isNatKind(M))
U61¹(tt, V1, V2) → ACTIVATE(V2)
U42¹(tt, V2) → ACTIVATE(V2)
U52¹(tt, V2) → ACTIVATE(V2)
U41¹(tt, V1, V2) → ISNAT(activate(V1))
U21¹(tt, V1) → ISNAT(activate(V1))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
TAKE(s(M), cons(N, IL)) → AND(isNatIList(activate(IL)), n__isNatIListKind(activate(IL)))
TAKE(0, IL) → AND(isNatIList(IL), n__isNatIListKind(IL))
U51¹(tt, V1, V2) → ACTIVATE(V1)
U91¹(tt, IL, M, N) → CONS(activate(N), n__take(activate(M), activate(IL)))
TAKE(s(M), cons(N, IL)) → AND(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))))
ISNATILIST(n__cons(V1, V2)) → U41¹(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U91¹(tt, IL, M, N) → ACTIVATE(M)
LENGTH(cons(N, L)) → AND(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N)))
TAKE(s(M), cons(N, IL)) → ISNAT(N)
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V1)
U11¹(tt, V1) → ACTIVATE(V1)
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V2)
U41¹(tt, V1, V2) → ACTIVATE(V1)
U51¹(tt, V1, V2) → U52¹(isNat(activate(V1)), activate(V2))
U21¹(tt, V1) → ACTIVATE(V1)
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
U42¹(tt, V2) → U43¹(isNatIList(activate(V2)))
U71¹(tt, L) → ACTIVATE(L)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ZEROS → CONS(0, n__zeros)
U91¹(tt, IL, M, N) → ACTIVATE(IL)
U11¹(tt, V1) → ISNATLIST(activate(V1))
U51¹(tt, V1, V2) → ACTIVATE(V2)
U71¹(tt, L) → LENGTH(activate(L))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
U61¹(tt, V1, V2) → ISNAT(activate(V1))
U52¹(tt, V2) → ISNATLIST(activate(V2))
U62¹(tt, V2) → ISNATILIST(activate(V2))
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V2)
U91¹(tt, IL, M, N) → ACTIVATE(N)
U42¹(tt, V2) → ISNATILIST(activate(V2))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__cons(X1, X2)) → CONS(X1, X2)
ISNATILISTKIND(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATILISTKIND(n__take(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ISNATLIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
LENGTH(nil) → 0¹
ACTIVATE(n__zeros) → ZEROS
U81¹(tt) → NIL
U52¹(tt, V2) → U53¹(isNatList(activate(V2)))
LENGTH(cons(N, L)) → U71¹(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
ACTIVATE(n__nil) → NIL
U61¹(tt, V1, V2) → ACTIVATE(V1)
U61¹(tt, V1, V2) → U62¹(isNat(activate(V1)), activate(V2))
ACTIVATE(n__s(X)) → S(X)
ISNAT(n__s(V1)) → ACTIVATE(V1)
AND(tt, X) → ACTIVATE(X)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
ISNATILIST(V) → ISNATILISTKIND(activate(V))
ISNATLIST(n__take(V1, V2)) → ISNATKIND(activate(V1))
ISNAT(n__s(V1)) → U21¹(isNatKind(activate(V1)), activate(V1))
U31¹(tt, V) → ISNATLIST(activate(V))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ISNATLIST(n__take(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V1)
ISNATILISTKIND(n__take(V1, V2)) → ACTIVATE(V2)
U41¹(tt, V1, V2) → U42¹(isNat(activate(V1)), activate(V2))
U21¹(tt, V1) → U22¹(isNat(activate(V1)))
ISNAT(n__length(V1)) → ISNATILISTKIND(activate(V1))
U31¹(tt, V) → ACTIVATE(V)
U31¹(tt, V) → U32¹(isNatList(activate(V)))
TAKE(0, IL) → U81¹(and(isNatIList(IL), n__isNatIListKind(IL)))
TAKE(s(M), cons(N, IL)) → ISNAT(M)
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
U41¹(tt, V1, V2) → ACTIVATE(V2)
U51¹(tt, V1, V2) → ISNAT(activate(V1))
LENGTH(cons(N, L)) → AND(isNatList(activate(L)), n__isNatIListKind(activate(L)))
LENGTH(cons(N, L)) → ACTIVATE(L)
ISNATLIST(n__cons(V1, V2)) → U51¹(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
ISNATLIST(n__take(V1, V2)) → U61¹(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
TAKE(s(M), cons(N, IL)) → U91¹(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ISNATILIST(V) → U31¹(isNatIListKind(activate(V)), activate(V))
U71¹(tt, L) → S(length(activate(L)))
U62¹(tt, V2) → ACTIVATE(V2)
ACTIVATE(n__take(X1, X2)) → TAKE(X1, X2)
ISNATILIST(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ISNAT(n__length(V1)) → ACTIVATE(V1)
ZEROS → 0¹

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, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
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)))
isNatIListKind(n__take(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))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 18 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

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

U11¹(tt, V1) → ISNATLIST(activate(V1))
ISNATILIST(V) → ACTIVATE(V)
U51¹(tt, V1, V2) → ACTIVATE(V2)
ISNATLIST(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
U71¹(tt, L) → LENGTH(activate(L))
ISNATILISTKIND(n__take(V1, V2)) → ACTIVATE(V1)
U61¹(tt, V1, V2) → ISNAT(activate(V1))
U52¹(tt, V2) → ISNATLIST(activate(V2))
ISNATILISTKIND(n__take(V1, V2)) → ISNATKIND(activate(V1))
U62¹(tt, V2) → ISNATILIST(activate(V2))
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V2)
U91¹(tt, IL, M, N) → ACTIVATE(N)
ISNATILISTKIND(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
U42¹(tt, V2) → ISNATILIST(activate(V2))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILISTKIND(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__length(V1)) → ACTIVATE(V1)
ISNATILISTKIND(n__take(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
LENGTH(cons(N, L)) → ISNAT(N)
ISNATILIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATLIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNAT(n__length(V1)) → U11¹(isNatIListKind(activate(V1)), activate(V1))
ISNATKIND(n__length(V1)) → ISNATILISTKIND(activate(V1))
ACTIVATE(n__length(X)) → LENGTH(X)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
LENGTH(cons(N, L)) → U71¹(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
TAKE(0, IL) → ISNATILIST(IL)
U61¹(tt, V1, V2) → ACTIVATE(V1)
U61¹(tt, V1, V2) → U62¹(isNat(activate(V1)), activate(V2))
ISNAT(n__s(V1)) → ACTIVATE(V1)
AND(tt, X) → ACTIVATE(X)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
TAKE(s(M), cons(N, IL)) → AND(isNat(M), n__isNatKind(M))
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
U61¹(tt, V1, V2) → ACTIVATE(V2)
U52¹(tt, V2) → ACTIVATE(V2)
U42¹(tt, V2) → ACTIVATE(V2)
ISNATILIST(V) → ISNATILISTKIND(activate(V))
ISNATLIST(n__take(V1, V2)) → ISNATKIND(activate(V1))
U41¹(tt, V1, V2) → ISNAT(activate(V1))
U21¹(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → U21¹(isNatKind(activate(V1)), activate(V1))
U31¹(tt, V) → ISNATLIST(activate(V))
TAKE(s(M), cons(N, IL)) → AND(isNatIList(activate(IL)), n__isNatIListKind(activate(IL)))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ISNATLIST(n__take(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V1)
ISNATILISTKIND(n__take(V1, V2)) → ACTIVATE(V2)
TAKE(0, IL) → AND(isNatIList(IL), n__isNatIListKind(IL))
U41¹(tt, V1, V2) → U42¹(isNat(activate(V1)), activate(V2))
U51¹(tt, V1, V2) → ACTIVATE(V1)
ISNAT(n__length(V1)) → ISNATILISTKIND(activate(V1))
U31¹(tt, V) → ACTIVATE(V)
TAKE(s(M), cons(N, IL)) → AND(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))))
TAKE(s(M), cons(N, IL)) → ISNAT(M)
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → U41¹(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U91¹(tt, IL, M, N) → ACTIVATE(M)
LENGTH(cons(N, L)) → AND(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N)))
TAKE(s(M), cons(N, IL)) → ISNAT(N)
U41¹(tt, V1, V2) → ACTIVATE(V2)
U51¹(tt, V1, V2) → ISNAT(activate(V1))
LENGTH(cons(N, L)) → AND(isNatList(activate(L)), n__isNatIListKind(activate(L)))
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V1)
U11¹(tt, V1) → ACTIVATE(V1)
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V2)
LENGTH(cons(N, L)) → ACTIVATE(L)
U41¹(tt, V1, V2) → 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))
U21¹(tt, V1) → ACTIVATE(V1)
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
TAKE(s(M), cons(N, IL)) → U91¹(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
ISNATLIST(n__take(V1, V2)) → U61¹(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U71¹(tt, L) → ACTIVATE(L)
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATILIST(V) → U31¹(isNatIListKind(activate(V)), activate(V))
U91¹(tt, IL, M, N) → ACTIVATE(IL)
U62¹(tt, V2) → ACTIVATE(V2)
ACTIVATE(n__take(X1, X2)) → TAKE(X1, X2)
ISNATILIST(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ISNAT(n__length(V1)) → ACTIVATE(V1)

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, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
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)))
isNatIListKind(n__take(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))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.

U11¹(tt, V1) → ISNATLIST(activate(V1))
ISNATILIST(V) → ACTIVATE(V)
U51¹(tt, V1, V2) → ACTIVATE(V2)
ISNATLIST(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
U71¹(tt, L) → LENGTH(activate(L))
ISNATILISTKIND(n__take(V1, V2)) → ACTIVATE(V1)
U61¹(tt, V1, V2) → ISNAT(activate(V1))
U52¹(tt, V2) → ISNATLIST(activate(V2))
ISNATILISTKIND(n__take(V1, V2)) → ISNATKIND(activate(V1))
U62¹(tt, V2) → ISNATILIST(activate(V2))
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V2)
U91¹(tt, IL, M, N) → ACTIVATE(N)
ISNATILISTKIND(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
U42¹(tt, V2) → ISNATILIST(activate(V2))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILISTKIND(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATILISTKIND(n__take(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
LENGTH(cons(N, L)) → ISNAT(N)
ISNATILIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATLIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
LENGTH(cons(N, L)) → U71¹(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
TAKE(0, IL) → ISNATILIST(IL)
U61¹(tt, V1, V2) → ACTIVATE(V1)
U61¹(tt, V1, V2) → U62¹(isNat(activate(V1)), activate(V2))
ISNAT(n__s(V1)) → ACTIVATE(V1)
AND(tt, X) → ACTIVATE(X)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
TAKE(s(M), cons(N, IL)) → AND(isNat(M), n__isNatKind(M))
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
U61¹(tt, V1, V2) → ACTIVATE(V2)
U52¹(tt, V2) → ACTIVATE(V2)
U42¹(tt, V2) → ACTIVATE(V2)
ISNATILIST(V) → ISNATILISTKIND(activate(V))
ISNATLIST(n__take(V1, V2)) → ISNATKIND(activate(V1))
U41¹(tt, V1, V2) → ISNAT(activate(V1))
U21¹(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → U21¹(isNatKind(activate(V1)), activate(V1))
U31¹(tt, V) → ISNATLIST(activate(V))
TAKE(s(M), cons(N, IL)) → AND(isNatIList(activate(IL)), n__isNatIListKind(activate(IL)))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ISNATLIST(n__take(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V1)
ISNATILISTKIND(n__take(V1, V2)) → ACTIVATE(V2)
TAKE(0, IL) → AND(isNatIList(IL), n__isNatIListKind(IL))
U41¹(tt, V1, V2) → U42¹(isNat(activate(V1)), activate(V2))
U51¹(tt, V1, V2) → ACTIVATE(V1)
U31¹(tt, V) → ACTIVATE(V)
TAKE(s(M), cons(N, IL)) → AND(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))))
TAKE(s(M), cons(N, IL)) → ISNAT(M)
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → U41¹(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U91¹(tt, IL, M, N) → ACTIVATE(M)
LENGTH(cons(N, L)) → AND(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N)))
TAKE(s(M), cons(N, IL)) → ISNAT(N)
U41¹(tt, V1, V2) → ACTIVATE(V2)
U51¹(tt, V1, V2) → ISNAT(activate(V1))
LENGTH(cons(N, L)) → AND(isNatList(activate(L)), n__isNatIListKind(activate(L)))
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V1)
U11¹(tt, V1) → ACTIVATE(V1)
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V2)
LENGTH(cons(N, L)) → ACTIVATE(L)
U41¹(tt, V1, V2) → 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))
U21¹(tt, V1) → ACTIVATE(V1)
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
TAKE(s(M), cons(N, IL)) → U91¹(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
ISNATLIST(n__take(V1, V2)) → U61¹(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U71¹(tt, L) → ACTIVATE(L)
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATILIST(V) → U31¹(isNatIListKind(activate(V)), activate(V))
U91¹(tt, IL, M, N) → ACTIVATE(IL)
U62¹(tt, V2) → ACTIVATE(V2)
ACTIVATE(n__take(X1, X2)) → TAKE(X1, X2)
ISNATILIST(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(ACTIVATE(x₁)) = x₁
POL(AND(x₁, x₂)) = x₂
POL(ISNAT(x₁)) = x₁
POL(ISNATILIST(x₁)) = x₁
POL(ISNATILISTKIND(x₁)) = x₁
POL(ISNATKIND(x₁)) = x₁
POL(ISNATLIST(x₁)) = x₁
POL(LENGTH(x₁)) = x₁
POL(TAKE(x₁, x₂)) = x₁ + x₂
POL(U11(x₁, x₂)) = 0
POL(U11¹(x₁, x₂)) = x₂
POL(U12(x₁)) = 0
POL(U21(x₁, x₂)) = 0
POL(U21¹(x₁, x₂)) = x₂
POL(U22(x₁)) = 0
POL(U31(x₁, x₂)) = 0
POL(U31¹(x₁, x₂)) = x₂
POL(U32(x₁)) = 0
POL(U41(x₁, x₂, x₃)) = 1
POL(U41¹(x₁, x₂, x₃)) = x₂ + x₃
POL(U42(x₁, x₂)) = 0
POL(U42¹(x₁, x₂)) = x₂
POL(U43(x₁)) = 0
POL(U51(x₁, x₂, x₃)) = 1 + x₃
POL(U51¹(x₁, x₂, x₃)) = x₂ + x₃
POL(U52(x₁, x₂)) = 1 + x₂
POL(U52¹(x₁, x₂)) = x₂
POL(U53(x₁)) = 0
POL(U61(x₁, x₂, x₃)) = 1
POL(U61¹(x₁, x₂, x₃)) = x₂ + x₃
POL(U62(x₁, x₂)) = 1
POL(U62¹(x₁, x₂)) = x₂
POL(U63(x₁)) = 1
POL(U71(x₁, x₂)) = 1 + x₂
POL(U71¹(x₁, x₂)) = x₂
POL(U81(x₁)) = x₁
POL(U91(x₁, x₂, x₃, x₄)) = x₂ + x₃ + x₄
POL(U91¹(x₁, x₂, x₃, x₄)) = x₂ + x₃ + x₄
POL(activate(x₁)) = x₁
POL(and(x₁, x₂)) = x₂
POL(cons(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 1
POL(isNatIListKind(x₁)) = x₁
POL(isNatKind(x₁)) = x₁
POL(isNatList(x₁)) = 1 + x₁
POL(length(x₁)) = 1 + x₁
POL(n__0) = 0
POL(n__and(x₁, x₂)) = x₂
POL(n__cons(x₁, x₂)) = x₁ + x₂
POL(n__isNatIListKind(x₁)) = x₁
POL(n__isNatKind(x₁)) = x₁
POL(n__length(x₁)) = 1 + x₁
POL(n__nil) = 0
POL(n__s(x₁)) = x₁
POL(n__take(x₁, x₂)) = x₁ + x₂
POL(n__zeros) = 0
POL(nil) = 0
POL(s(x₁)) = x₁
POL(take(x₁, x₂)) = x₁ + x₂
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

zeros → cons(0, n__zeros)
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U81(tt) → nil
U62(tt, V2) → U63(isNatIList(activate(V2)))
U61(tt, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U71(tt, L) → s(length(activate(L)))
U63(tt) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U43(tt) → tt
U53(tt) → tt
U52(tt, V2) → U53(isNatList(activate(V2)))
U32(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U42(tt, V2) → U43(isNatIList(activate(V2)))
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
isNatKind(n__0) → 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
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))
activate(n__0) → 0
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__zeros) → zeros
isNatKind(X) → n__isNatKind(X)
and(X1, X2) → n__and(X1, X2)
nil → n__nil
isNatIListKind(X) → n__isNatIListKind(X)
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(X) → n__length(X)
0 → n__0
take(X1, X2) → n__take(X1, X2)
zeros → n__zeros
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
activate(X) → X
activate(n__nil) → nil
activate(n__cons(X1, X2)) → cons(X1, X2)
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
activate(n__isNatIListKind(X)) → isNatIListKind(X)
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
and(tt, X) → activate(X)
activate(n__isNatKind(X)) → isNatKind(X)
isNatIListKind(n__take(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

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

U11¹(tt, V1) → ISNATLIST(activate(V1))
ISNATILIST(V) → ACTIVATE(V)
U51¹(tt, V1, V2) → ACTIVATE(V2)
ISNATLIST(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
U71¹(tt, L) → LENGTH(activate(L))
ISNATILISTKIND(n__take(V1, V2)) → ACTIVATE(V1)
U61¹(tt, V1, V2) → ISNAT(activate(V1))
U52¹(tt, V2) → ISNATLIST(activate(V2))
ISNATILISTKIND(n__take(V1, V2)) → ISNATKIND(activate(V1))
U62¹(tt, V2) → ISNATILIST(activate(V2))
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V2)
ISNATILISTKIND(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
U91¹(tt, IL, M, N) → ACTIVATE(N)
U42¹(tt, V2) → ISNATILIST(activate(V2))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILISTKIND(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATILISTKIND(n__take(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ISNATILIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
LENGTH(cons(N, L)) → ISNAT(N)
ISNATLIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
LENGTH(cons(N, L)) → U71¹(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
TAKE(0, IL) → ISNATILIST(IL)
U61¹(tt, V1, V2) → ACTIVATE(V1)
U61¹(tt, V1, V2) → U62¹(isNat(activate(V1)), activate(V2))
ISNAT(n__s(V1)) → ACTIVATE(V1)
AND(tt, X) → ACTIVATE(X)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
TAKE(s(M), cons(N, IL)) → AND(isNat(M), n__isNatKind(M))
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
U61¹(tt, V1, V2) → ACTIVATE(V2)
U52¹(tt, V2) → ACTIVATE(V2)
U42¹(tt, V2) → ACTIVATE(V2)
ISNATILIST(V) → ISNATILISTKIND(activate(V))
ISNATLIST(n__take(V1, V2)) → ISNATKIND(activate(V1))
U41¹(tt, V1, V2) → ISNAT(activate(V1))
U21¹(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → U21¹(isNatKind(activate(V1)), activate(V1))
U31¹(tt, V) → ISNATLIST(activate(V))
TAKE(s(M), cons(N, IL)) → AND(isNatIList(activate(IL)), n__isNatIListKind(activate(IL)))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ISNATLIST(n__take(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V1)
ISNATILISTKIND(n__take(V1, V2)) → ACTIVATE(V2)
TAKE(0, IL) → AND(isNatIList(IL), n__isNatIListKind(IL))
U41¹(tt, V1, V2) → U42¹(isNat(activate(V1)), activate(V2))
U51¹(tt, V1, V2) → ACTIVATE(V1)
U31¹(tt, V) → ACTIVATE(V)
TAKE(s(M), cons(N, IL)) → AND(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))))
TAKE(s(M), cons(N, IL)) → ISNAT(M)
ISNATILIST(n__cons(V1, V2)) → U41¹(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
LENGTH(cons(N, L)) → AND(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N)))
U91¹(tt, IL, M, N) → ACTIVATE(M)
TAKE(s(M), cons(N, IL)) → ISNAT(N)
U41¹(tt, V1, V2) → ACTIVATE(V2)
LENGTH(cons(N, L)) → AND(isNatList(activate(L)), n__isNatIListKind(activate(L)))
U51¹(tt, V1, V2) → ISNAT(activate(V1))
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V1)
U11¹(tt, V1) → ACTIVATE(V1)
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V2)
LENGTH(cons(N, L)) → ACTIVATE(L)
U41¹(tt, V1, V2) → 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))
U21¹(tt, V1) → ACTIVATE(V1)
ISNATLIST(n__take(V1, V2)) → U61¹(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
TAKE(s(M), cons(N, IL)) → U91¹(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
U71¹(tt, L) → ACTIVATE(L)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATILIST(V) → U31¹(isNatIListKind(activate(V)), activate(V))
ACTIVATE(n__take(X1, X2)) → TAKE(X1, X2)
U62¹(tt, V2) → ACTIVATE(V2)
U91¹(tt, IL, M, N) → ACTIVATE(IL)
ISNATILIST(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(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, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
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)))
isNatIListKind(n__take(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))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs with 8 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                  ↳ QDP

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

ISNATILIST(V) → ACTIVATE(V)
U51¹(tt, V1, V2) → ACTIVATE(V2)
ISNATLIST(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATILISTKIND(n__take(V1, V2)) → ACTIVATE(V1)
U52¹(tt, V2) → ISNATLIST(activate(V2))
U61¹(tt, V1, V2) → ISNAT(activate(V1))
ISNATILISTKIND(n__take(V1, V2)) → ISNATKIND(activate(V1))
U62¹(tt, V2) → ISNATILIST(activate(V2))
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V2)
U91¹(tt, IL, M, N) → ACTIVATE(N)
ISNATILISTKIND(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
U42¹(tt, V2) → ISNATILIST(activate(V2))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILISTKIND(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATILISTKIND(n__take(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ISNATILIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATLIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
U61¹(tt, V1, V2) → ACTIVATE(V1)
TAKE(0, IL) → ISNATILIST(IL)
U61¹(tt, V1, V2) → U62¹(isNat(activate(V1)), activate(V2))
AND(tt, X) → ACTIVATE(X)
ISNAT(n__s(V1)) → ACTIVATE(V1)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
TAKE(s(M), cons(N, IL)) → AND(isNat(M), n__isNatKind(M))
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
U61¹(tt, V1, V2) → ACTIVATE(V2)
U52¹(tt, V2) → ACTIVATE(V2)
ISNATILIST(V) → ISNATILISTKIND(activate(V))
U42¹(tt, V2) → ACTIVATE(V2)
ISNATLIST(n__take(V1, V2)) → ISNATKIND(activate(V1))
U41¹(tt, V1, V2) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → U21¹(isNatKind(activate(V1)), activate(V1))
U21¹(tt, V1) → ISNAT(activate(V1))
U31¹(tt, V) → ISNATLIST(activate(V))
TAKE(s(M), cons(N, IL)) → AND(isNatIList(activate(IL)), n__isNatIListKind(activate(IL)))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V1)
ISNATLIST(n__take(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ISNATILISTKIND(n__take(V1, V2)) → ACTIVATE(V2)
TAKE(0, IL) → AND(isNatIList(IL), n__isNatIListKind(IL))
U41¹(tt, V1, V2) → U42¹(isNat(activate(V1)), activate(V2))
U51¹(tt, V1, V2) → ACTIVATE(V1)
U31¹(tt, V) → ACTIVATE(V)
TAKE(s(M), cons(N, IL)) → AND(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))))
TAKE(s(M), cons(N, IL)) → ISNAT(M)
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → U41¹(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U91¹(tt, IL, M, N) → ACTIVATE(M)
TAKE(s(M), cons(N, IL)) → ISNAT(N)
U41¹(tt, V1, V2) → ACTIVATE(V2)
U51¹(tt, V1, V2) → ISNAT(activate(V1))
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V2)
U41¹(tt, V1, V2) → 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))
U21¹(tt, V1) → ACTIVATE(V1)
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
TAKE(s(M), cons(N, IL)) → U91¹(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
ISNATLIST(n__take(V1, V2)) → U61¹(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATILIST(V) → U31¹(isNatIListKind(activate(V)), activate(V))
U91¹(tt, IL, M, N) → ACTIVATE(IL)
ACTIVATE(n__take(X1, X2)) → TAKE(X1, X2)
U62¹(tt, V2) → ACTIVATE(V2)
ISNATILIST(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(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, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
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)))
isNatIListKind(n__take(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))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

ISNATILIST(V) → ACTIVATE(V)
U51¹(tt, V1, V2) → ACTIVATE(V2)
ISNATLIST(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATILISTKIND(n__take(V1, V2)) → ACTIVATE(V1)
U61¹(tt, V1, V2) → ISNAT(activate(V1))
ISNATILISTKIND(n__take(V1, V2)) → ISNATKIND(activate(V1))
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V2)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILISTKIND(n__take(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ISNATILIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATLIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
U61¹(tt, V1, V2) → ACTIVATE(V1)
TAKE(s(M), cons(N, IL)) → AND(isNat(M), n__isNatKind(M))
U61¹(tt, V1, V2) → ACTIVATE(V2)
U52¹(tt, V2) → ACTIVATE(V2)
ISNATILIST(V) → ISNATILISTKIND(activate(V))
U42¹(tt, V2) → ACTIVATE(V2)
ISNATLIST(n__take(V1, V2)) → ISNATKIND(activate(V1))
U41¹(tt, V1, V2) → ISNAT(activate(V1))
TAKE(s(M), cons(N, IL)) → AND(isNatIList(activate(IL)), n__isNatIListKind(activate(IL)))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATLIST(n__take(V1, V2)) → ACTIVATE(V1)
ISNATLIST(n__take(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ISNATILISTKIND(n__take(V1, V2)) → ACTIVATE(V2)
TAKE(0, IL) → AND(isNatIList(IL), n__isNatIListKind(IL))
U51¹(tt, V1, V2) → ACTIVATE(V1)
U31¹(tt, V) → ACTIVATE(V)
TAKE(s(M), cons(N, IL)) → AND(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))))
TAKE(s(M), cons(N, IL)) → ISNAT(M)
TAKE(s(M), cons(N, IL)) → ISNAT(N)
U41¹(tt, V1, V2) → ACTIVATE(V2)
U51¹(tt, V1, V2) → ISNAT(activate(V1))
U41¹(tt, V1, V2) → ACTIVATE(V1)
TAKE(s(M), cons(N, IL)) → ACTIVATE(IL)
TAKE(s(M), cons(N, IL)) → U91¹(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
ISNATLIST(n__take(V1, V2)) → U61¹(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U62¹(tt, V2) → ACTIVATE(V2)
ISNATILIST(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))

The remaining pairs can at least be oriented weakly.

U52¹(tt, V2) → ISNATLIST(activate(V2))
U62¹(tt, V2) → ISNATILIST(activate(V2))
U91¹(tt, IL, M, N) → ACTIVATE(N)
ISNATILISTKIND(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
U42¹(tt, V2) → ISNATILIST(activate(V2))
ISNATILISTKIND(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
TAKE(0, IL) → ISNATILIST(IL)
U61¹(tt, V1, V2) → U62¹(isNat(activate(V1)), activate(V2))
AND(tt, X) → ACTIVATE(X)
ISNAT(n__s(V1)) → ACTIVATE(V1)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
ISNAT(n__s(V1)) → U21¹(isNatKind(activate(V1)), activate(V1))
U21¹(tt, V1) → ISNAT(activate(V1))
U31¹(tt, V) → ISNATLIST(activate(V))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
U41¹(tt, V1, V2) → U42¹(isNat(activate(V1)), activate(V2))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → U41¹(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U91¹(tt, IL, M, N) → ACTIVATE(M)
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V2)
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))
U21¹(tt, V1) → ACTIVATE(V1)
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATILIST(V) → U31¹(isNatIListKind(activate(V)), activate(V))
U91¹(tt, IL, M, N) → ACTIVATE(IL)
ACTIVATE(n__take(X1, X2)) → TAKE(X1, X2)

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(ACTIVATE(x₁)) = x₁
POL(AND(x₁, x₂)) = x₂
POL(ISNAT(x₁)) = x₁
POL(ISNATILIST(x₁)) = 1 + x₁
POL(ISNATILISTKIND(x₁)) = x₁
POL(ISNATKIND(x₁)) = x₁
POL(ISNATLIST(x₁)) = 1 + x₁
POL(TAKE(x₁, x₂)) = 1 + x₁ + x₂
POL(U11(x₁, x₂)) = 0
POL(U12(x₁)) = 0
POL(U21(x₁, x₂)) = 0
POL(U21¹(x₁, x₂)) = x₂
POL(U22(x₁)) = 0
POL(U31(x₁, x₂)) = 0
POL(U31¹(x₁, x₂)) = 1 + x₂
POL(U32(x₁)) = 0
POL(U41(x₁, x₂, x₃)) = 0
POL(U41¹(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(U42(x₁, x₂)) = 0
POL(U42¹(x₁, x₂)) = 1 + x₂
POL(U43(x₁)) = 0
POL(U51(x₁, x₂, x₃)) = 0
POL(U51¹(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(U52(x₁, x₂)) = 0
POL(U52¹(x₁, x₂)) = 1 + x₂
POL(U53(x₁)) = 0
POL(U61(x₁, x₂, x₃)) = 0
POL(U61¹(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(U62(x₁, x₂)) = 0
POL(U62¹(x₁, x₂)) = 1 + x₂
POL(U63(x₁)) = 0
POL(U71(x₁, x₂)) = x₂
POL(U81(x₁)) = 1 + x₁
POL(U91(x₁, x₂, x₃, x₄)) = 1 + x₂ + x₃ + x₄
POL(U91¹(x₁, x₂, x₃, x₄)) = x₂ + x₃ + x₄
POL(activate(x₁)) = x₁
POL(and(x₁, x₂)) = x₂
POL(cons(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatIListKind(x₁)) = x₁
POL(isNatKind(x₁)) = x₁
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = x₁
POL(n__0) = 0
POL(n__and(x₁, x₂)) = x₂
POL(n__cons(x₁, x₂)) = x₁ + x₂
POL(n__isNatIListKind(x₁)) = x₁
POL(n__isNatKind(x₁)) = x₁
POL(n__length(x₁)) = x₁
POL(n__nil) = 1
POL(n__s(x₁)) = x₁
POL(n__take(x₁, x₂)) = 1 + x₁ + x₂
POL(n__zeros) = 0
POL(nil) = 1
POL(s(x₁)) = x₁
POL(take(x₁, x₂)) = 1 + x₁ + x₂
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

zeros → cons(0, n__zeros)
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U81(tt) → nil
U62(tt, V2) → U63(isNatIList(activate(V2)))
U61(tt, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U71(tt, L) → s(length(activate(L)))
U63(tt) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U43(tt) → tt
U53(tt) → tt
U52(tt, V2) → U53(isNatList(activate(V2)))
U32(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U42(tt, V2) → U43(isNatIList(activate(V2)))
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
isNatKind(n__0) → 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
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))
activate(n__0) → 0
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__zeros) → zeros
isNatKind(X) → n__isNatKind(X)
and(X1, X2) → n__and(X1, X2)
nil → n__nil
isNatIListKind(X) → n__isNatIListKind(X)
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(X) → n__length(X)
0 → n__0
take(X1, X2) → n__take(X1, X2)
zeros → n__zeros
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
activate(X) → X
activate(n__nil) → nil
activate(n__cons(X1, X2)) → cons(X1, X2)
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
activate(n__isNatIListKind(X)) → isNatIListKind(X)
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
and(tt, X) → activate(X)
activate(n__isNatKind(X)) → isNatKind(X)
isNatIListKind(n__take(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                  ↳ QDP

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

ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
U41¹(tt, V1, V2) → U42¹(isNat(activate(V1)), activate(V2))
U52¹(tt, V2) → ISNATLIST(activate(V2))
U62¹(tt, V2) → ISNATILIST(activate(V2))
ISNATILISTKIND(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
U91¹(tt, IL, M, N) → ACTIVATE(N)
U42¹(tt, V2) → ISNATILIST(activate(V2))
ISNATILISTKIND(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → U41¹(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
U91¹(tt, IL, M, N) → ACTIVATE(M)
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATLIST(n__cons(V1, V2)) → U51¹(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U61¹(tt, V1, V2) → U62¹(isNat(activate(V1)), activate(V2))
TAKE(0, IL) → ISNATILIST(IL)
U51¹(tt, V1, V2) → U52¹(isNat(activate(V1)), activate(V2))
ISNAT(n__s(V1)) → ACTIVATE(V1)
AND(tt, X) → ACTIVATE(X)
TAKE(s(M), cons(N, IL)) → ISNATILIST(activate(IL))
U21¹(tt, V1) → ACTIVATE(V1)
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATILIST(V) → U31¹(isNatIListKind(activate(V)), activate(V))
ISNAT(n__s(V1)) → U21¹(isNatKind(activate(V1)), activate(V1))
U21¹(tt, V1) → ISNAT(activate(V1))
ACTIVATE(n__take(X1, X2)) → TAKE(X1, X2)
U91¹(tt, IL, M, N) → ACTIVATE(IL)
U31¹(tt, V) → ISNATLIST(activate(V))

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, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
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)))
isNatIListKind(n__take(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))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 4 SCCs with 13 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ AND

                            ↳ QDP

                              ↳ Narrowing

                            ↳ QDP

                            ↳ QDP

                            ↳ QDP

                  ↳ QDP

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

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))

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, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
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)))
isNatIListKind(n__take(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))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U51¹(tt, V1, V2) → U52¹(isNat(activate(V1)), activate(V2)) at position [0] we obtained the following new rules:

U51¹(tt, n__zeros, y1) → U52¹(isNat(zeros), activate(y1))
U51¹(tt, n__s(x0), y1) → U52¹(isNat(s(x0)), activate(y1))
U51¹(tt, n__isNatKind(x0), y1) → U52¹(isNat(isNatKind(x0)), activate(y1))
U51¹(tt, n__cons(x0, x1), y1) → U52¹(isNat(cons(x0, x1)), activate(y1))
U51¹(tt, n__0, y1) → U52¹(isNat(0), activate(y1))
U51¹(tt, n__length(x0), y1) → U52¹(isNat(length(x0)), activate(y1))
U51¹(tt, n__and(x0, x1), y1) → U52¹(isNat(and(x0, x1)), activate(y1))
U51¹(tt, x0, y1) → U52¹(isNat(x0), activate(y1))
U51¹(tt, n__nil, y1) → U52¹(isNat(nil), activate(y1))
U51¹(tt, n__take(x0, x1), y1) → U52¹(isNat(take(x0, x1)), activate(y1))
U51¹(tt, n__isNatIListKind(x0), y1) → U52¹(isNat(isNatIListKind(x0)), activate(y1))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ AND

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                            ↳ QDP

                            ↳ QDP

                            ↳ QDP

                  ↳ QDP

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

U51¹(tt, n__zeros, y1) → U52¹(isNat(zeros), activate(y1))
U51¹(tt, n__cons(x0, x1), y1) → U52¹(isNat(cons(x0, x1)), activate(y1))
U51¹(tt, n__0, y1) → U52¹(isNat(0), activate(y1))
U51¹(tt, n__length(x0), y1) → U52¹(isNat(length(x0)), activate(y1))
U51¹(tt, x0, y1) → U52¹(isNat(x0), activate(y1))
U51¹(tt, n__nil, y1) → U52¹(isNat(nil), activate(y1))
ISNATLIST(n__cons(V1, V2)) → U51¹(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U52¹(tt, V2) → ISNATLIST(activate(V2))
U51¹(tt, n__isNatIListKind(x0), y1) → U52¹(isNat(isNatIListKind(x0)), activate(y1))
U51¹(tt, n__s(x0), y1) → U52¹(isNat(s(x0)), activate(y1))
U51¹(tt, n__isNatKind(x0), y1) → U52¹(isNat(isNatKind(x0)), activate(y1))
U51¹(tt, n__and(x0, x1), y1) → U52¹(isNat(and(x0, x1)), activate(y1))
U51¹(tt, n__take(x0, x1), y1) → U52¹(isNat(take(x0, x1)), activate(y1))

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, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
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)))
isNatIListKind(n__take(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))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U52¹(tt, V2) → ISNATLIST(activate(V2)) at position [0] we obtained the following new rules:

U52¹(tt, x0) → ISNATLIST(x0)
U52¹(tt, n__s(x0)) → ISNATLIST(s(x0))
U52¹(tt, n__nil) → ISNATLIST(nil)
U52¹(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
U52¹(tt, n__isNatKind(x0)) → ISNATLIST(isNatKind(x0))
U52¹(tt, n__0) → ISNATLIST(0)
U52¹(tt, n__zeros) → ISNATLIST(zeros)
U52¹(tt, n__length(x0)) → ISNATLIST(length(x0))
U52¹(tt, n__cons(x0, x1)) → ISNATLIST(cons(x0, x1))
U52¹(tt, n__isNatIListKind(x0)) → ISNATLIST(isNatIListKind(x0))
U52¹(tt, n__and(x0, x1)) → ISNATLIST(and(x0, x1))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ AND

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                            ↳ QDP

                            ↳ QDP

                            ↳ QDP

                  ↳ QDP

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

U51¹(tt, n__zeros, y1) → U52¹(isNat(zeros), activate(y1))
U51¹(tt, n__cons(x0, x1), y1) → U52¹(isNat(cons(x0, x1)), activate(y1))
U51¹(tt, n__0, y1) → U52¹(isNat(0), activate(y1))
U51¹(tt, n__length(x0), y1) → U52¹(isNat(length(x0)), activate(y1))
U51¹(tt, n__nil, y1) → U52¹(isNat(nil), activate(y1))
U51¹(tt, x0, y1) → U52¹(isNat(x0), activate(y1))
ISNATLIST(n__cons(V1, V2)) → U51¹(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U52¹(tt, n__isNatKind(x0)) → ISNATLIST(isNatKind(x0))
U51¹(tt, n__isNatIListKind(x0), y1) → U52¹(isNat(isNatIListKind(x0)), activate(y1))
U52¹(tt, x0) → ISNATLIST(x0)
U51¹(tt, n__s(x0), y1) → U52¹(isNat(s(x0)), activate(y1))
U51¹(tt, n__isNatKind(x0), y1) → U52¹(isNat(isNatKind(x0)), activate(y1))
U51¹(tt, n__and(x0, x1), y1) → U52¹(isNat(and(x0, x1)), activate(y1))
U52¹(tt, n__s(x0)) → ISNATLIST(s(x0))
U52¹(tt, n__nil) → ISNATLIST(nil)
U52¹(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
U51¹(tt, n__take(x0, x1), y1) → U52¹(isNat(take(x0, x1)), activate(y1))
U52¹(tt, n__0) → ISNATLIST(0)
U52¹(tt, n__zeros) → ISNATLIST(zeros)
U52¹(tt, n__length(x0)) → ISNATLIST(length(x0))
U52¹(tt, n__isNatIListKind(x0)) → ISNATLIST(isNatIListKind(x0))
U52¹(tt, n__cons(x0, x1)) → ISNATLIST(cons(x0, x1))
U52¹(tt, n__and(x0, x1)) → ISNATLIST(and(x0, x1))

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, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
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)))
isNatIListKind(n__take(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))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U51¹(tt, n__zeros, y1) → U52¹(isNat(zeros), activate(y1)) at position [0] we obtained the following new rules:

U51¹(tt, n__zeros, y0) → U52¹(isNat(cons(0, n__zeros)), activate(y0))
U51¹(tt, n__zeros, y0) → U52¹(isNat(n__zeros), activate(y0))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ AND

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                            ↳ QDP

                            ↳ QDP

                            ↳ QDP

                  ↳ QDP

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

U51¹(tt, n__length(x0), y1) → U52¹(isNat(length(x0)), activate(y1))
U51¹(tt, n__isNatIListKind(x0), y1) → U52¹(isNat(isNatIListKind(x0)), activate(y1))
U51¹(tt, n__zeros, y0) → U52¹(isNat(n__zeros), activate(y0))
U52¹(tt, x0) → ISNATLIST(x0)
U51¹(tt, n__s(x0), y1) → U52¹(isNat(s(x0)), activate(y1))
U52¹(tt, n__nil) → ISNATLIST(nil)
U52¹(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
U52¹(tt, n__zeros) → ISNATLIST(zeros)
U52¹(tt, n__length(x0)) → ISNATLIST(length(x0))
U51¹(tt, n__cons(x0, x1), y1) → U52¹(isNat(cons(x0, x1)), activate(y1))
U51¹(tt, n__0, y1) → U52¹(isNat(0), activate(y1))
U51¹(tt, n__nil, y1) → U52¹(isNat(nil), activate(y1))
U51¹(tt, x0, y1) → U52¹(isNat(x0), activate(y1))
ISNATLIST(n__cons(V1, V2)) → U51¹(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U52¹(tt, n__isNatKind(x0)) → ISNATLIST(isNatKind(x0))
U51¹(tt, n__isNatKind(x0), y1) → U52¹(isNat(isNatKind(x0)), activate(y1))
U51¹(tt, n__and(x0, x1), y1) → U52¹(isNat(and(x0, x1)), activate(y1))
U52¹(tt, n__s(x0)) → ISNATLIST(s(x0))
U51¹(tt, n__take(x0, x1), y1) → U52¹(isNat(take(x0, x1)), activate(y1))
U52¹(tt, n__0) → ISNATLIST(0)
U51¹(tt, n__zeros, y0) → U52¹(isNat(cons(0, n__zeros)), activate(y0))
U52¹(tt, n__isNatIListKind(x0)) → ISNATLIST(isNatIListKind(x0))
U52¹(tt, n__cons(x0, x1)) → ISNATLIST(cons(x0, x1))
U52¹(tt, n__and(x0, x1)) → ISNATLIST(and(x0, x1))

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, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
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)))
isNatIListKind(n__take(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))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ AND

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                            ↳ QDP

                            ↳ QDP

                            ↳ QDP

                  ↳ QDP

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

U51¹(tt, n__cons(x0, x1), y1) → U52¹(isNat(cons(x0, x1)), activate(y1))
U51¹(tt, n__0, y1) → U52¹(isNat(0), activate(y1))
U51¹(tt, n__length(x0), y1) → U52¹(isNat(length(x0)), activate(y1))
U51¹(tt, n__nil, y1) → U52¹(isNat(nil), activate(y1))
U51¹(tt, x0, y1) → U52¹(isNat(x0), activate(y1))
ISNATLIST(n__cons(V1, V2)) → U51¹(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U52¹(tt, n__isNatKind(x0)) → ISNATLIST(isNatKind(x0))
U51¹(tt, n__isNatIListKind(x0), y1) → U52¹(isNat(isNatIListKind(x0)), activate(y1))
U51¹(tt, n__s(x0), y1) → U52¹(isNat(s(x0)), activate(y1))
U52¹(tt, x0) → ISNATLIST(x0)
U51¹(tt, n__isNatKind(x0), y1) → U52¹(isNat(isNatKind(x0)), activate(y1))
U51¹(tt, n__and(x0, x1), y1) → U52¹(isNat(and(x0, x1)), activate(y1))
U52¹(tt, n__s(x0)) → ISNATLIST(s(x0))
U52¹(tt, n__nil) → ISNATLIST(nil)
U51¹(tt, n__take(x0, x1), y1) → U52¹(isNat(take(x0, x1)), activate(y1))
U52¹(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
U52¹(tt, n__0) → ISNATLIST(0)
U52¹(tt, n__zeros) → ISNATLIST(zeros)
U52¹(tt, n__length(x0)) → ISNATLIST(length(x0))
U51¹(tt, n__zeros, y0) → U52¹(isNat(cons(0, n__zeros)), activate(y0))
U52¹(tt, n__cons(x0, x1)) → ISNATLIST(cons(x0, x1))
U52¹(tt, n__isNatIListKind(x0)) → ISNATLIST(isNatIListKind(x0))
U52¹(tt, n__and(x0, x1)) → ISNATLIST(and(x0, x1))

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, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
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)))
isNatIListKind(n__take(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))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U51¹(tt, n__s(x0), y1) → U52¹(isNat(s(x0)), activate(y1)) at position [0] we obtained the following new rules:

U51¹(tt, n__s(x0), y1) → U52¹(isNat(n__s(x0)), activate(y1))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ AND

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                            ↳ QDP

                            ↳ QDP

                            ↳ QDP

                  ↳ QDP

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

U51¹(tt, n__cons(x0, x1), y1) → U52¹(isNat(cons(x0, x1)), activate(y1))
U51¹(tt, n__0, y1) → U52¹(isNat(0), activate(y1))
U51¹(tt, n__length(x0), y1) → U52¹(isNat(length(x0)), activate(y1))
U51¹(tt, x0, y1) → U52¹(isNat(x0), activate(y1))
U51¹(tt, n__nil, y1) → U52¹(isNat(nil), activate(y1))
ISNATLIST(n__cons(V1, V2)) → U51¹(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U52¹(tt, n__isNatKind(x0)) → ISNATLIST(isNatKind(x0))
U51¹(tt, n__s(x0), y1) → U52¹(isNat(n__s(x0)), activate(y1))
U51¹(tt, n__isNatIListKind(x0), y1) → U52¹(isNat(isNatIListKind(x0)), activate(y1))
U52¹(tt, x0) → ISNATLIST(x0)
U51¹(tt, n__isNatKind(x0), y1) → U52¹(isNat(isNatKind(x0)), activate(y1))
U51¹(tt, n__and(x0, x1), y1) → U52¹(isNat(and(x0, x1)), activate(y1))
U52¹(tt, n__s(x0)) → ISNATLIST(s(x0))
U52¹(tt, n__nil) → ISNATLIST(nil)
U52¹(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
U51¹(tt, n__take(x0, x1), y1) → U52¹(isNat(take(x0, x1)), activate(y1))
U52¹(tt, n__0) → ISNATLIST(0)
U52¹(tt, n__zeros) → ISNATLIST(zeros)
U52¹(tt, n__length(x0)) → ISNATLIST(length(x0))
U51¹(tt, n__zeros, y0) → U52¹(isNat(cons(0, n__zeros)), activate(y0))
U52¹(tt, n__isNatIListKind(x0)) → ISNATLIST(isNatIListKind(x0))
U52¹(tt, n__cons(x0, x1)) → ISNATLIST(cons(x0, x1))
U52¹(tt, n__and(x0, x1)) → ISNATLIST(and(x0, x1))

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, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
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)))
isNatIListKind(n__take(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))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U52¹(tt, n__s(x0)) → ISNATLIST(s(x0)) at position [0] we obtained the following new rules:

U52¹(tt, n__s(x0)) → ISNATLIST(n__s(x0))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ AND

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                            ↳ QDP

                            ↳ QDP

                            ↳ QDP

                  ↳ QDP

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

U51¹(tt, n__cons(x0, x1), y1) → U52¹(isNat(cons(x0, x1)), activate(y1))
U51¹(tt, n__0, y1) → U52¹(isNat(0), activate(y1))
U51¹(tt, n__length(x0), y1) → U52¹(isNat(length(x0)), activate(y1))
U51¹(tt, n__nil, y1) → U52¹(isNat(nil), activate(y1))
U51¹(tt, x0, y1) → U52¹(isNat(x0), activate(y1))
ISNATLIST(n__cons(V1, V2)) → U51¹(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U52¹(tt, n__isNatKind(x0)) → ISNATLIST(isNatKind(x0))
U51¹(tt, n__s(x0), y1) → U52¹(isNat(n__s(x0)), activate(y1))
U51¹(tt, n__isNatIListKind(x0), y1) → U52¹(isNat(isNatIListKind(x0)), activate(y1))
U52¹(tt, x0) → ISNATLIST(x0)
U52¹(tt, n__s(x0)) → ISNATLIST(n__s(x0))
U51¹(tt, n__isNatKind(x0), y1) → U52¹(isNat(isNatKind(x0)), activate(y1))
U51¹(tt, n__and(x0, x1), y1) → U52¹(isNat(and(x0, x1)), activate(y1))
U52¹(tt, n__nil) → ISNATLIST(nil)
U51¹(tt, n__take(x0, x1), y1) → U52¹(isNat(take(x0, x1)), activate(y1))
U52¹(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
U52¹(tt, n__0) → ISNATLIST(0)
U52¹(tt, n__zeros) → ISNATLIST(zeros)
U52¹(tt, n__length(x0)) → ISNATLIST(length(x0))
U51¹(tt, n__zeros, y0) → U52¹(isNat(cons(0, n__zeros)), activate(y0))
U52¹(tt, n__cons(x0, x1)) → ISNATLIST(cons(x0, x1))
U52¹(tt, n__isNatIListKind(x0)) → ISNATLIST(isNatIListKind(x0))
U52¹(tt, n__and(x0, x1)) → ISNATLIST(and(x0, x1))

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, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
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)))
isNatIListKind(n__take(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))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ AND

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                            ↳ QDP

                            ↳ QDP

                            ↳ QDP

                  ↳ QDP

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

U51¹(tt, n__cons(x0, x1), y1) → U52¹(isNat(cons(x0, x1)), activate(y1))
U51¹(tt, n__0, y1) → U52¹(isNat(0), activate(y1))
U51¹(tt, n__length(x0), y1) → U52¹(isNat(length(x0)), activate(y1))
U51¹(tt, n__nil, y1) → U52¹(isNat(nil), activate(y1))
U51¹(tt, x0, y1) → U52¹(isNat(x0), activate(y1))
ISNATLIST(n__cons(V1, V2)) → U51¹(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U52¹(tt, n__isNatKind(x0)) → ISNATLIST(isNatKind(x0))
U51¹(tt, n__s(x0), y1) → U52¹(isNat(n__s(x0)), activate(y1))
U51¹(tt, n__isNatIListKind(x0), y1) → U52¹(isNat(isNatIListKind(x0)), activate(y1))
U52¹(tt, x0) → ISNATLIST(x0)
U51¹(tt, n__isNatKind(x0), y1) → U52¹(isNat(isNatKind(x0)), activate(y1))
U51¹(tt, n__and(x0, x1), y1) → U52¹(isNat(and(x0, x1)), activate(y1))
U52¹(tt, n__nil) → ISNATLIST(nil)
U51¹(tt, n__take(x0, x1), y1) → U52¹(isNat(take(x0, x1)), activate(y1))
U52¹(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
U52¹(tt, n__0) → ISNATLIST(0)
U52¹(tt, n__zeros) → ISNATLIST(zeros)
U52¹(tt, n__length(x0)) → ISNATLIST(length(x0))
U51¹(tt, n__zeros, y0) → U52¹(isNat(cons(0, n__zeros)), activate(y0))
U52¹(tt, n__cons(x0, x1)) → ISNATLIST(cons(x0, x1))
U52¹(tt, n__isNatIListKind(x0)) → ISNATLIST(isNatIListKind(x0))
U52¹(tt, n__and(x0, x1)) → ISNATLIST(and(x0, x1))

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, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
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)))
isNatIListKind(n__take(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))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U52¹(tt, n__nil) → ISNATLIST(nil) at position [0] we obtained the following new rules:

U52¹(tt, n__nil) → ISNATLIST(n__nil)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ AND

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                            ↳ QDP

                            ↳ QDP

                            ↳ QDP

                  ↳ QDP

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

U51¹(tt, n__cons(x0, x1), y1) → U52¹(isNat(cons(x0, x1)), activate(y1))
U51¹(tt, n__0, y1) → U52¹(isNat(0), activate(y1))
U51¹(tt, n__length(x0), y1) → U52¹(isNat(length(x0)), activate(y1))
U51¹(tt, x0, y1) → U52¹(isNat(x0), activate(y1))
U51¹(tt, n__nil, y1) → U52¹(isNat(nil), activate(y1))
ISNATLIST(n__cons(V1, V2)) → U51¹(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U52¹(tt, n__isNatKind(x0)) → ISNATLIST(isNatKind(x0))
U51¹(tt, n__s(x0), y1) → U52¹(isNat(n__s(x0)), activate(y1))
U51¹(tt, n__isNatIListKind(x0), y1) → U52¹(isNat(isNatIListKind(x0)), activate(y1))
U52¹(tt, x0) → ISNATLIST(x0)
U51¹(tt, n__isNatKind(x0), y1) → U52¹(isNat(isNatKind(x0)), activate(y1))
U52¹(tt, n__nil) → ISNATLIST(n__nil)
U51¹(tt, n__and(x0, x1), y1) → U52¹(isNat(and(x0, x1)), activate(y1))
U52¹(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
U51¹(tt, n__take(x0, x1), y1) → U52¹(isNat(take(x0, x1)), activate(y1))
U52¹(tt, n__0) → ISNATLIST(0)
U52¹(tt, n__zeros) → ISNATLIST(zeros)
U52¹(tt, n__length(x0)) → ISNATLIST(length(x0))
U51¹(tt, n__zeros, y0) → U52¹(isNat(cons(0, n__zeros)), activate(y0))
U52¹(tt, n__isNatIListKind(x0)) → ISNATLIST(isNatIListKind(x0))
U52¹(tt, n__cons(x0, x1)) → ISNATLIST(cons(x0, x1))
U52¹(tt, n__and(x0, x1)) → ISNATLIST(and(x0, x1))

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, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
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)))
isNatIListKind(n__take(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))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ AND

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                            ↳ QDP

                            ↳ QDP

                            ↳ QDP

                  ↳ QDP

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

U51¹(tt, n__cons(x0, x1), y1) → U52¹(isNat(cons(x0, x1)), activate(y1))
U51¹(tt, n__0, y1) → U52¹(isNat(0), activate(y1))
U51¹(tt, n__length(x0), y1) → U52¹(isNat(length(x0)), activate(y1))
U51¹(tt, n__nil, y1) → U52¹(isNat(nil), activate(y1))
U51¹(tt, x0, y1) → U52¹(isNat(x0), activate(y1))
ISNATLIST(n__cons(V1, V2)) → U51¹(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U52¹(tt, n__isNatKind(x0)) → ISNATLIST(isNatKind(x0))
U51¹(tt, n__s(x0), y1) → U52¹(isNat(n__s(x0)), activate(y1))
U51¹(tt, n__isNatIListKind(x0), y1) → U52¹(isNat(isNatIListKind(x0)), activate(y1))
U52¹(tt, x0) → ISNATLIST(x0)
U51¹(tt, n__isNatKind(x0), y1) → U52¹(isNat(isNatKind(x0)), activate(y1))
U51¹(tt, n__and(x0, x1), y1) → U52¹(isNat(and(x0, x1)), activate(y1))
U51¹(tt, n__take(x0, x1), y1) → U52¹(isNat(take(x0, x1)), activate(y1))
U52¹(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
U52¹(tt, n__0) → ISNATLIST(0)
U52¹(tt, n__zeros) → ISNATLIST(zeros)
U52¹(tt, n__length(x0)) → ISNATLIST(length(x0))
U51¹(tt, n__zeros, y0) → U52¹(isNat(cons(0, n__zeros)), activate(y0))
U52¹(tt, n__cons(x0, x1)) → ISNATLIST(cons(x0, x1))
U52¹(tt, n__isNatIListKind(x0)) → ISNATLIST(isNatIListKind(x0))
U52¹(tt, n__and(x0, x1)) → ISNATLIST(and(x0, x1))

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, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
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)))
isNatIListKind(n__take(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))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U51¹(tt, n__cons(x0, x1), y1) → U52¹(isNat(cons(x0, x1)), activate(y1)) at position [0] we obtained the following new rules:

U51¹(tt, n__cons(x0, x1), y2) → U52¹(isNat(n__cons(x0, x1)), activate(y2))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ AND

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                            ↳ QDP

                            ↳ QDP

                            ↳ QDP

                  ↳ QDP

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

U51¹(tt, n__0, y1) → U52¹(isNat(0), activate(y1))
U51¹(tt, n__length(x0), y1) → U52¹(isNat(length(x0)), activate(y1))
U51¹(tt, x0, y1) → U52¹(isNat(x0), activate(y1))
U51¹(tt, n__nil, y1) → U52¹(isNat(nil), activate(y1))
ISNATLIST(n__cons(V1, V2)) → U51¹(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U52¹(tt, n__isNatKind(x0)) → ISNATLIST(isNatKind(x0))
U51¹(tt, n__s(x0), y1) → U52¹(isNat(n__s(x0)), activate(y1))
U51¹(tt, n__isNatIListKind(x0), y1) → U52¹(isNat(isNatIListKind(x0)), activate(y1))
U52¹(tt, x0) → ISNATLIST(x0)
U51¹(tt, n__isNatKind(x0), y1) → U52¹(isNat(isNatKind(x0)), activate(y1))
U51¹(tt, n__and(x0, x1), y1) → U52¹(isNat(and(x0, x1)), activate(y1))
U52¹(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
U51¹(tt, n__take(x0, x1), y1) → U52¹(isNat(take(x0, x1)), activate(y1))
U52¹(tt, n__0) → ISNATLIST(0)
U52¹(tt, n__zeros) → ISNATLIST(zeros)
U52¹(tt, n__length(x0)) → ISNATLIST(length(x0))
U51¹(tt, n__cons(x0, x1), y2) → U52¹(isNat(n__cons(x0, x1)), activate(y2))
U51¹(tt, n__zeros, y0) → U52¹(isNat(cons(0, n__zeros)), activate(y0))
U52¹(tt, n__isNatIListKind(x0)) → ISNATLIST(isNatIListKind(x0))
U52¹(tt, n__cons(x0, x1)) → ISNATLIST(cons(x0, x1))
U52¹(tt, n__and(x0, x1)) → ISNATLIST(and(x0, x1))

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, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
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)))
isNatIListKind(n__take(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))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ AND

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                            ↳ QDP

                            ↳ QDP

                            ↳ QDP

                  ↳ QDP

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

U51¹(tt, n__0, y1) → U52¹(isNat(0), activate(y1))
U51¹(tt, n__length(x0), y1) → U52¹(isNat(length(x0)), activate(y1))
U51¹(tt, n__nil, y1) → U52¹(isNat(nil), activate(y1))
U51¹(tt, x0, y1) → U52¹(isNat(x0), activate(y1))
ISNATLIST(n__cons(V1, V2)) → U51¹(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U52¹(tt, n__isNatKind(x0)) → ISNATLIST(isNatKind(x0))
U51¹(tt, n__s(x0), y1) → U52¹(isNat(n__s(x0)), activate(y1))
U51¹(tt, n__isNatIListKind(x0), y1) → U52¹(isNat(isNatIListKind(x0)), activate(y1))
U52¹(tt, x0) → ISNATLIST(x0)
U51¹(tt, n__isNatKind(x0), y1) → U52¹(isNat(isNatKind(x0)), activate(y1))
U51¹(tt, n__and(x0, x1), y1) → U52¹(isNat(and(x0, x1)), activate(y1))
U51¹(tt, n__take(x0, x1), y1) → U52¹(isNat(take(x0, x1)), activate(y1))
U52¹(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
U52¹(tt, n__0) → ISNATLIST(0)
U52¹(tt, n__zeros) → ISNATLIST(zeros)
U52¹(tt, n__length(x0)) → ISNATLIST(length(x0))
U51¹(tt, n__zeros, y0) → U52¹(isNat(cons(0, n__zeros)), activate(y0))
U52¹(tt, n__cons(x0, x1)) → ISNATLIST(cons(x0, x1))
U52¹(tt, n__isNatIListKind(x0)) → ISNATLIST(isNatIListKind(x0))
U52¹(tt, n__and(x0, x1)) → ISNATLIST(and(x0, x1))

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, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
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)))
isNatIListKind(n__take(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))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U52¹(tt, n__0) → ISNATLIST(0) at position [0] we obtained the following new rules:

U52¹(tt, n__0) → ISNATLIST(n__0)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ AND

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                            ↳ QDP

                            ↳ QDP

                            ↳ QDP

                  ↳ QDP

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

U51¹(tt, n__0, y1) → U52¹(isNat(0), activate(y1))
U51¹(tt, n__length(x0), y1) → U52¹(isNat(length(x0)), activate(y1))
U51¹(tt, x0, y1) → U52¹(isNat(x0), activate(y1))
U51¹(tt, n__nil, y1) → U52¹(isNat(nil), activate(y1))
ISNATLIST(n__cons(V1, V2)) → U51¹(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U52¹(tt, n__isNatKind(x0)) → ISNATLIST(isNatKind(x0))
U51¹(tt, n__s(x0), y1) → U52¹(isNat(n__s(x0)), activate(y1))
U51¹(tt, n__isNatIListKind(x0), y1) → U52¹(isNat(isNatIListKind(x0)), activate(y1))
U52¹(tt, n__0) → ISNATLIST(n__0)
U52¹(tt, x0) → ISNATLIST(x0)
U51¹(tt, n__isNatKind(x0), y1) → U52¹(isNat(isNatKind(x0)), activate(y1))
U51¹(tt, n__and(x0, x1), y1) → U52¹(isNat(and(x0, x1)), activate(y1))
U52¹(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
U51¹(tt, n__take(x0, x1), y1) → U52¹(isNat(take(x0, x1)), activate(y1))
U52¹(tt, n__zeros) → ISNATLIST(zeros)
U52¹(tt, n__length(x0)) → ISNATLIST(length(x0))
U51¹(tt, n__zeros, y0) → U52¹(isNat(cons(0, n__zeros)), activate(y0))
U52¹(tt, n__isNatIListKind(x0)) → ISNATLIST(isNatIListKind(x0))
U52¹(tt, n__cons(x0, x1)) → ISNATLIST(cons(x0, x1))
U52¹(tt, n__and(x0, x1)) → ISNATLIST(and(x0, x1))

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, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
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)))
isNatIListKind(n__take(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))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ AND

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                            ↳ QDP

                            ↳ QDP

                            ↳ QDP

                  ↳ QDP

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

U51¹(tt, n__0, y1) → U52¹(isNat(0), activate(y1))
U51¹(tt, n__length(x0), y1) → U52¹(isNat(length(x0)), activate(y1))
U51¹(tt, n__nil, y1) → U52¹(isNat(nil), activate(y1))
U51¹(tt, x0, y1) → U52¹(isNat(x0), activate(y1))
ISNATLIST(n__cons(V1, V2)) → U51¹(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U52¹(tt, n__isNatKind(x0)) → ISNATLIST(isNatKind(x0))
U51¹(tt, n__s(x0), y1) → U52¹(isNat(n__s(x0)), activate(y1))
U51¹(tt, n__isNatIListKind(x0), y1) → U52¹(isNat(isNatIListKind(x0)), activate(y1))
U52¹(tt, x0) → ISNATLIST(x0)
U51¹(tt, n__isNatKind(x0), y1) → U52¹(isNat(isNatKind(x0)), activate(y1))
U51¹(tt, n__and(x0, x1), y1) → U52¹(isNat(and(x0, x1)), activate(y1))
U51¹(tt, n__take(x0, x1), y1) → U52¹(isNat(take(x0, x1)), activate(y1))
U52¹(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
U52¹(tt, n__zeros) → ISNATLIST(zeros)
U52¹(tt, n__length(x0)) → ISNATLIST(length(x0))
U51¹(tt, n__zeros, y0) → U52¹(isNat(cons(0, n__zeros)), activate(y0))
U52¹(tt, n__cons(x0, x1)) → ISNATLIST(cons(x0, x1))
U52¹(tt, n__isNatIListKind(x0)) → ISNATLIST(isNatIListKind(x0))
U52¹(tt, n__and(x0, x1)) → ISNATLIST(and(x0, x1))

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, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
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)))
isNatIListKind(n__take(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))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U52¹(tt, n__zeros) → ISNATLIST(zeros) at position [0] we obtained the following new rules:

U52¹(tt, n__zeros) → ISNATLIST(cons(0, n__zeros))
U52¹(tt, n__zeros) → ISNATLIST(n__zeros)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ AND

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                            ↳ QDP

                            ↳ QDP

                            ↳ QDP

                  ↳ QDP

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

U51¹(tt, n__0, y1) → U52¹(isNat(0), activate(y1))
U51¹(tt, n__length(x0), y1) → U52¹(isNat(length(x0)), activate(y1))
U52¹(tt, n__zeros) → ISNATLIST(n__zeros)
U51¹(tt, x0, y1) → U52¹(isNat(x0), activate(y1))
U51¹(tt, n__nil, y1) → U52¹(isNat(nil), activate(y1))
ISNATLIST(n__cons(V1, V2)) → U51¹(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U52¹(tt, n__isNatKind(x0)) → ISNATLIST(isNatKind(x0))
U51¹(tt, n__s(x0), y1) → U52¹(isNat(n__s(x0)), activate(y1))
U51¹(tt, n__isNatIListKind(x0), y1) → U52¹(isNat(isNatIListKind(x0)), activate(y1))
U52¹(tt, x0) → ISNATLIST(x0)
U51¹(tt, n__isNatKind(x0), y1) → U52¹(isNat(isNatKind(x0)), activate(y1))
U52¹(tt, n__zeros) → ISNATLIST(cons(0, n__zeros))
U51¹(tt, n__and(x0, x1), y1) → U52¹(isNat(and(x0, x1)), activate(y1))
U52¹(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
U51¹(tt, n__take(x0, x1), y1) → U52¹(isNat(take(x0, x1)), activate(y1))
U52¹(tt, n__length(x0)) → ISNATLIST(length(x0))
U51¹(tt, n__zeros, y0) → U52¹(isNat(cons(0, n__zeros)), activate(y0))
U52¹(tt, n__isNatIListKind(x0)) → ISNATLIST(isNatIListKind(x0))
U52¹(tt, n__cons(x0, x1)) → ISNATLIST(cons(x0, x1))
U52¹(tt, n__and(x0, x1)) → ISNATLIST(and(x0, x1))

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, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
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)))
isNatIListKind(n__take(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))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ AND

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                            ↳ QDP

                            ↳ QDP

                            ↳ QDP

                  ↳ QDP

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

U51¹(tt, n__0, y1) → U52¹(isNat(0), activate(y1))
U51¹(tt, n__length(x0), y1) → U52¹(isNat(length(x0)), activate(y1))
U51¹(tt, n__nil, y1) → U52¹(isNat(nil), activate(y1))
U51¹(tt, x0, y1) → U52¹(isNat(x0), activate(y1))
ISNATLIST(n__cons(V1, V2)) → U51¹(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U52¹(tt, n__isNatKind(x0)) → ISNATLIST(isNatKind(x0))
U51¹(tt, n__s(x0), y1) → U52¹(isNat(n__s(x0)), activate(y1))
U51¹(tt, n__isNatIListKind(x0), y1) → U52¹(isNat(isNatIListKind(x0)), activate(y1))
U52¹(tt, x0) → ISNATLIST(x0)
U51¹(tt, n__isNatKind(x0), y1) → U52¹(isNat(isNatKind(x0)), activate(y1))
U52¹(tt, n__zeros) → ISNATLIST(cons(0, n__zeros))
U51¹(tt, n__and(x0, x1), y1) → U52¹(isNat(and(x0, x1)), activate(y1))
U51¹(tt, n__take(x0, x1), y1) → U52¹(isNat(take(x0, x1)), activate(y1))
U52¹(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
U52¹(tt, n__length(x0)) → ISNATLIST(length(x0))
U51¹(tt, n__zeros, y0) → U52¹(isNat(cons(0, n__zeros)), activate(y0))
U52¹(tt, n__cons(x0, x1)) → ISNATLIST(cons(x0, x1))
U52¹(tt, n__isNatIListKind(x0)) → ISNATLIST(isNatIListKind(x0))
U52¹(tt, n__and(x0, x1)) → ISNATLIST(and(x0, x1))

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, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
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)))
isNatIListKind(n__take(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))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U51¹(tt, n__0, y1) → U52¹(isNat(0), activate(y1)) at position [0] we obtained the following new rules:

U51¹(tt, n__0, y0) → U52¹(isNat(n__0), activate(y0))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ AND

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                            ↳ QDP

                            ↳ QDP

                            ↳ QDP

                  ↳ QDP

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

U51¹(tt, n__length(x0), y1) → U52¹(isNat(length(x0)), activate(y1))
U51¹(tt, x0, y1) → U52¹(isNat(x0), activate(y1))
U51¹(tt, n__nil, y1) → U52¹(isNat(nil), activate(y1))
ISNATLIST(n__cons(V1, V2)) → U51¹(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U52¹(tt, n__isNatKind(x0)) → ISNATLIST(isNatKind(x0))
U51¹(tt, n__s(x0), y1) → U52¹(isNat(n__s(x0)), activate(y1))
U51¹(tt, n__0, y0) → U52¹(isNat(n__0), activate(y0))
U51¹(tt, n__isNatIListKind(x0), y1) → U52¹(isNat(isNatIListKind(x0)), activate(y1))
U52¹(tt, x0) → ISNATLIST(x0)
U51¹(tt, n__isNatKind(x0), y1) → U52¹(isNat(isNatKind(x0)), activate(y1))
U51¹(tt, n__and(x0, x1), y1) → U52¹(isNat(and(x0, x1)), activate(y1))
U52¹(tt, n__zeros) → ISNATLIST(cons(0, n__zeros))
U52¹(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
U51¹(tt, n__take(x0, x1), y1) → U52¹(isNat(take(x0, x1)), activate(y1))
U52¹(tt, n__length(x0)) → ISNATLIST(length(x0))
U51¹(tt, n__zeros, y0) → U52¹(isNat(cons(0, n__zeros)), activate(y0))
U52¹(tt, n__isNatIListKind(x0)) → ISNATLIST(isNatIListKind(x0))
U52¹(tt, n__cons(x0, x1)) → ISNATLIST(cons(x0, x1))
U52¹(tt, n__and(x0, x1)) → ISNATLIST(and(x0, x1))

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, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
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)))
isNatIListKind(n__take(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))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U52¹(tt, n__cons(x0, x1)) → ISNATLIST(cons(x0, x1)) at position [0] we obtained the following new rules:

U52¹(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ AND

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                            ↳ QDP

                            ↳ QDP

                            ↳ QDP

                  ↳ QDP

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

U51¹(tt, n__length(x0), y1) → U52¹(isNat(length(x0)), activate(y1))
U51¹(tt, n__nil, y1) → U52¹(isNat(nil), activate(y1))
U51¹(tt, x0, y1) → U52¹(isNat(x0), activate(y1))
ISNATLIST(n__cons(V1, V2)) → U51¹(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U52¹(tt, n__isNatKind(x0)) → ISNATLIST(isNatKind(x0))
U51¹(tt, n__s(x0), y1) → U52¹(isNat(n__s(x0)), activate(y1))
U51¹(tt, n__0, y0) → U52¹(isNat(n__0), activate(y0))
U51¹(tt, n__isNatIListKind(x0), y1) → U52¹(isNat(isNatIListKind(x0)), activate(y1))
U52¹(tt, x0) → ISNATLIST(x0)
U51¹(tt, n__isNatKind(x0), y1) → U52¹(isNat(isNatKind(x0)), activate(y1))
U52¹(tt, n__zeros) → ISNATLIST(cons(0, n__zeros))
U51¹(tt, n__and(x0, x1), y1) → U52¹(isNat(and(x0, x1)), activate(y1))
U52¹(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
U51¹(tt, n__take(x0, x1), y1) → U52¹(isNat(take(x0, x1)), activate(y1))
U52¹(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
U52¹(tt, n__length(x0)) → ISNATLIST(length(x0))
U51¹(tt, n__zeros, y0) → U52¹(isNat(cons(0, n__zeros)), activate(y0))
U52¹(tt, n__isNatIListKind(x0)) → ISNATLIST(isNatIListKind(x0))
U52¹(tt, n__and(x0, x1)) → ISNATLIST(and(x0, x1))

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, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
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)))
isNatIListKind(n__take(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))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U52¹(tt, n__zeros) → ISNATLIST(cons(0, n__zeros)) at position [0] we obtained the following new rules:

U52¹(tt, n__zeros) → ISNATLIST(cons(n__0, n__zeros))
U52¹(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ AND

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                            ↳ QDP

                            ↳ QDP

                            ↳ QDP

                  ↳ QDP

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

U52¹(tt, n__zeros) → ISNATLIST(cons(n__0, n__zeros))
U51¹(tt, n__length(x0), y1) → U52¹(isNat(length(x0)), activate(y1))
U51¹(tt, x0, y1) → U52¹(isNat(x0), activate(y1))
U51¹(tt, n__nil, y1) → U52¹(isNat(nil), activate(y1))
ISNATLIST(n__cons(V1, V2)) → U51¹(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U52¹(tt, n__isNatKind(x0)) → ISNATLIST(isNatKind(x0))
U51¹(tt, n__s(x0), y1) → U52¹(isNat(n__s(x0)), activate(y1))
U52¹(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
U51¹(tt, n__0, y0) → U52¹(isNat(n__0), activate(y0))
U51¹(tt, n__isNatIListKind(x0), y1) → U52¹(isNat(isNatIListKind(x0)), activate(y1))
U52¹(tt, x0) → ISNATLIST(x0)
U51¹(tt, n__isNatKind(x0), y1) → U52¹(isNat(isNatKind(x0)), activate(y1))
U51¹(tt, n__and(x0, x1), y1) → U52¹(isNat(and(x0, x1)), activate(y1))
U52¹(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
U52¹(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
U51¹(tt, n__take(x0, x1), y1) → U52¹(isNat(take(x0, x1)), activate(y1))
U52¹(tt, n__length(x0)) → ISNATLIST(length(x0))
U51¹(tt, n__zeros, y0) → U52¹(isNat(cons(0, n__zeros)), activate(y0))
U52¹(tt, n__isNatIListKind(x0)) → ISNATLIST(isNatIListKind(x0))
U52¹(tt, n__and(x0, x1)) → ISNATLIST(and(x0, x1))

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, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
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)))
isNatIListKind(n__take(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))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U51¹(tt, n__nil, y1) → U52¹(isNat(nil), activate(y1)) at position [0] we obtained the following new rules:

U51¹(tt, n__nil, y0) → U52¹(isNat(n__nil), activate(y0))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ AND

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                            ↳ QDP

                            ↳ QDP

                            ↳ QDP

                  ↳ QDP

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

U51¹(tt, n__nil, y0) → U52¹(isNat(n__nil), activate(y0))
U52¹(tt, n__zeros) → ISNATLIST(cons(n__0, n__zeros))
U51¹(tt, n__length(x0), y1) → U52¹(isNat(length(x0)), activate(y1))
U51¹(tt, x0, y1) → U52¹(isNat(x0), activate(y1))
ISNATLIST(n__cons(V1, V2)) → U51¹(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U52¹(tt, n__isNatKind(x0)) → ISNATLIST(isNatKind(x0))
U51¹(tt, n__s(x0), y1) → U52¹(isNat(n__s(x0)), activate(y1))
U51¹(tt, n__0, y0) → U52¹(isNat(n__0), activate(y0))
U52¹(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
U51¹(tt, n__isNatIListKind(x0), y1) → U52¹(isNat(isNatIListKind(x0)), activate(y1))
U52¹(tt, x0) → ISNATLIST(x0)
U51¹(tt, n__isNatKind(x0), y1) → U52¹(isNat(isNatKind(x0)), activate(y1))
U51¹(tt, n__and(x0, x1), y1) → U52¹(isNat(and(x0, x1)), activate(y1))
U52¹(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
U51¹(tt, n__take(x0, x1), y1) → U52¹(isNat(take(x0, x1)), activate(y1))
U52¹(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
U52¹(tt, n__length(x0)) → ISNATLIST(length(x0))
U51¹(tt, n__zeros, y0) → U52¹(isNat(cons(0, n__zeros)), activate(y0))
U52¹(tt, n__isNatIListKind(x0)) → ISNATLIST(isNatIListKind(x0))
U52¹(tt, n__and(x0, x1)) → ISNATLIST(and(x0, x1))

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, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
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)))
isNatIListKind(n__take(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))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ AND

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                            ↳ QDP

                            ↳ QDP

                            ↳ QDP

                  ↳ QDP

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

U52¹(tt, n__zeros) → ISNATLIST(cons(n__0, n__zeros))
U51¹(tt, n__length(x0), y1) → U52¹(isNat(length(x0)), activate(y1))
U51¹(tt, x0, y1) → U52¹(isNat(x0), activate(y1))
ISNATLIST(n__cons(V1, V2)) → U51¹(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U52¹(tt, n__isNatKind(x0)) → ISNATLIST(isNatKind(x0))
U51¹(tt, n__s(x0), y1) → U52¹(isNat(n__s(x0)), activate(y1))
U51¹(tt, n__0, y0) → U52¹(isNat(n__0), activate(y0))
U52¹(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
U51¹(tt, n__isNatIListKind(x0), y1) → U52¹(isNat(isNatIListKind(x0)), activate(y1))
U52¹(tt, x0) → ISNATLIST(x0)
U51¹(tt, n__isNatKind(x0), y1) → U52¹(isNat(isNatKind(x0)), activate(y1))
U51¹(tt, n__and(x0, x1), y1) → U52¹(isNat(and(x0, x1)), activate(y1))
U52¹(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
U51¹(tt, n__take(x0, x1), y1) → U52¹(isNat(take(x0, x1)), activate(y1))
U52¹(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
U52¹(tt, n__length(x0)) → ISNATLIST(length(x0))
U51¹(tt, n__zeros, y0) → U52¹(isNat(cons(0, n__zeros)), activate(y0))
U52¹(tt, n__isNatIListKind(x0)) → ISNATLIST(isNatIListKind(x0))
U52¹(tt, n__and(x0, x1)) → ISNATLIST(and(x0, x1))

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, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
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)))
isNatIListKind(n__take(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))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U52¹(tt, n__zeros) → ISNATLIST(cons(n__0, n__zeros)) at position [0] we obtained the following new rules:

U52¹(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ AND

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                            ↳ QDP

                            ↳ QDP

                            ↳ QDP

                  ↳ QDP

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

U51¹(tt, n__length(x0), y1) → U52¹(isNat(length(x0)), activate(y1))
U51¹(tt, x0, y1) → U52¹(isNat(x0), activate(y1))
ISNATLIST(n__cons(V1, V2)) → U51¹(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U52¹(tt, n__isNatKind(x0)) → ISNATLIST(isNatKind(x0))
U51¹(tt, n__s(x0), y1) → U52¹(isNat(n__s(x0)), activate(y1))
U52¹(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
U51¹(tt, n__0, y0) → U52¹(isNat(n__0), activate(y0))
U51¹(tt, n__isNatIListKind(x0), y1) → U52¹(isNat(isNatIListKind(x0)), activate(y1))
U52¹(tt, x0) → ISNATLIST(x0)
U51¹(tt, n__isNatKind(x0), y1) → U52¹(isNat(isNatKind(x0)), activate(y1))
U51¹(tt, n__and(x0, x1), y1) → U52¹(isNat(and(x0, x1)), activate(y1))
U52¹(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
U52¹(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
U51¹(tt, n__take(x0, x1), y1) → U52¹(isNat(take(x0, x1)), activate(y1))
U52¹(tt, n__length(x0)) → ISNATLIST(length(x0))
U51¹(tt, n__zeros, y0) → U52¹(isNat(cons(0, n__zeros)), activate(y0))
U52¹(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))
U52¹(tt, n__isNatIListKind(x0)) → ISNATLIST(isNatIListKind(x0))
U52¹(tt, n__and(x0, x1)) → ISNATLIST(and(x0, x1))

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, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
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)))
isNatIListKind(n__take(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))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U51¹(tt, n__zeros, y0) → U52¹(isNat(cons(0, n__zeros)), activate(y0)) at position [0] we obtained the following new rules:

U51¹(tt, n__zeros, y0) → U52¹(isNat(n__cons(0, n__zeros)), activate(y0))
U51¹(tt, n__zeros, y0) → U52¹(isNat(cons(n__0, n__zeros)), activate(y0))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ AND

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

                            ↳ QDP

                            ↳ QDP

                            ↳ QDP

                  ↳ QDP

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

U51¹(tt, n__zeros, y0) → U52¹(isNat(n__cons(0, n__zeros)), activate(y0))
U51¹(tt, n__length(x0), y1) → U52¹(isNat(length(x0)), activate(y1))
U51¹(tt, x0, y1) → U52¹(isNat(x0), activate(y1))
ISNATLIST(n__cons(V1, V2)) → U51¹(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U52¹(tt, n__isNatKind(x0)) → ISNATLIST(isNatKind(x0))
U51¹(tt, n__s(x0), y1) → U52¹(isNat(n__s(x0)), activate(y1))
U51¹(tt, n__0, y0) → U52¹(isNat(n__0), activate(y0))
U52¹(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
U51¹(tt, n__isNatIListKind(x0), y1) → U52¹(isNat(isNatIListKind(x0)), activate(y1))
U52¹(tt, x0) → ISNATLIST(x0)
U51¹(tt, n__isNatKind(x0), y1) → U52¹(isNat(isNatKind(x0)), activate(y1))
U51¹(tt, n__zeros, y0) → U52¹(isNat(cons(n__0, n__zeros)), activate(y0))
U51¹(tt, n__and(x0, x1), y1) → U52¹(isNat(and(x0, x1)), activate(y1))
U52¹(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
U51¹(tt, n__take(x0, x1), y1) → U52¹(isNat(take(x0, x1)), activate(y1))
U52¹(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
U52¹(tt, n__length(x0)) → ISNATLIST(length(x0))
U52¹(tt, n__isNatIListKind(x0)) → ISNATLIST(isNatIListKind(x0))
U52¹(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))
U52¹(tt, n__and(x0, x1)) → ISNATLIST(and(x0, x1))

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, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
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)))
isNatIListKind(n__take(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))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ AND

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ Narrowing

                            ↳ QDP

                            ↳ QDP

                            ↳ QDP

                  ↳ QDP

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

U51¹(tt, n__length(x0), y1) → U52¹(isNat(length(x0)), activate(y1))
U51¹(tt, x0, y1) → U52¹(isNat(x0), activate(y1))
ISNATLIST(n__cons(V1, V2)) → U51¹(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U52¹(tt, n__isNatKind(x0)) → ISNATLIST(isNatKind(x0))
U51¹(tt, n__s(x0), y1) → U52¹(isNat(n__s(x0)), activate(y1))
U51¹(tt, n__0, y0) → U52¹(isNat(n__0), activate(y0))
U52¹(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
U51¹(tt, n__isNatIListKind(x0), y1) → U52¹(isNat(isNatIListKind(x0)), activate(y1))
U52¹(tt, x0) → ISNATLIST(x0)
U51¹(tt, n__isNatKind(x0), y1) → U52¹(isNat(isNatKind(x0)), activate(y1))
U51¹(tt, n__zeros, y0) → U52¹(isNat(cons(n__0, n__zeros)), activate(y0))
U51¹(tt, n__and(x0, x1), y1) → U52¹(isNat(and(x0, x1)), activate(y1))
U52¹(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
U51¹(tt, n__take(x0, x1), y1) → U52¹(isNat(take(x0, x1)), activate(y1))
U52¹(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
U52¹(tt, n__length(x0)) → ISNATLIST(length(x0))
U52¹(tt, n__isNatIListKind(x0)) → ISNATLIST(isNatIListKind(x0))
U52¹(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))
U52¹(tt, n__and(x0, x1)) → ISNATLIST(and(x0, x1))

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, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
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)))
isNatIListKind(n__take(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))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U51¹(tt, n__zeros, y0) → U52¹(isNat(cons(n__0, n__zeros)), activate(y0)) at position [0] we obtained the following new rules:

U51¹(tt, n__zeros, y0) → U52¹(isNat(n__cons(n__0, n__zeros)), activate(y0))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ AND

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ Narrowing

                                                                                                                            ↳ QDP

                                                                                                                              ↳ DependencyGraphProof

                            ↳ QDP

                            ↳ QDP

                            ↳ QDP

                  ↳ QDP

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

U51¹(tt, n__length(x0), y1) → U52¹(isNat(length(x0)), activate(y1))
U51¹(tt, x0, y1) → U52¹(isNat(x0), activate(y1))
ISNATLIST(n__cons(V1, V2)) → U51¹(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U52¹(tt, n__isNatKind(x0)) → ISNATLIST(isNatKind(x0))
U51¹(tt, n__zeros, y0) → U52¹(isNat(n__cons(n__0, n__zeros)), activate(y0))
U51¹(tt, n__s(x0), y1) → U52¹(isNat(n__s(x0)), activate(y1))
U52¹(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
U51¹(tt, n__0, y0) → U52¹(isNat(n__0), activate(y0))
U51¹(tt, n__isNatIListKind(x0), y1) → U52¹(isNat(isNatIListKind(x0)), activate(y1))
U52¹(tt, x0) → ISNATLIST(x0)
U51¹(tt, n__isNatKind(x0), y1) → U52¹(isNat(isNatKind(x0)), activate(y1))
U51¹(tt, n__and(x0, x1), y1) → U52¹(isNat(and(x0, x1)), activate(y1))
U52¹(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
U52¹(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
U51¹(tt, n__take(x0, x1), y1) → U52¹(isNat(take(x0, x1)), activate(y1))
U52¹(tt, n__length(x0)) → ISNATLIST(length(x0))
U52¹(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))
U52¹(tt, n__isNatIListKind(x0)) → ISNATLIST(isNatIListKind(x0))
U52¹(tt, n__and(x0, x1)) → ISNATLIST(and(x0, x1))

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, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
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)))
isNatIListKind(n__take(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))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ AND

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ Narrowing

                                                                                                                            ↳ QDP

                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ QDPOrderProof

                            ↳ QDP

                            ↳ QDP

                            ↳ QDP

                  ↳ QDP

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

U51¹(tt, n__length(x0), y1) → U52¹(isNat(length(x0)), activate(y1))
U51¹(tt, x0, y1) → U52¹(isNat(x0), activate(y1))
ISNATLIST(n__cons(V1, V2)) → U51¹(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U52¹(tt, n__isNatKind(x0)) → ISNATLIST(isNatKind(x0))
U51¹(tt, n__s(x0), y1) → U52¹(isNat(n__s(x0)), activate(y1))
U51¹(tt, n__0, y0) → U52¹(isNat(n__0), activate(y0))
U52¹(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
U51¹(tt, n__isNatIListKind(x0), y1) → U52¹(isNat(isNatIListKind(x0)), activate(y1))
U52¹(tt, x0) → ISNATLIST(x0)
U51¹(tt, n__isNatKind(x0), y1) → U52¹(isNat(isNatKind(x0)), activate(y1))
U51¹(tt, n__and(x0, x1), y1) → U52¹(isNat(and(x0, x1)), activate(y1))
U52¹(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
U51¹(tt, n__take(x0, x1), y1) → U52¹(isNat(take(x0, x1)), activate(y1))
U52¹(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
U52¹(tt, n__length(x0)) → ISNATLIST(length(x0))
U52¹(tt, n__isNatIListKind(x0)) → ISNATLIST(isNatIListKind(x0))
U52¹(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))
U52¹(tt, n__and(x0, x1)) → ISNATLIST(and(x0, x1))

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, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
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)))
isNatIListKind(n__take(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))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

U51¹(tt, n__take(x0, x1), y1) → U52¹(isNat(take(x0, x1)), activate(y1))

The remaining pairs can at least be oriented weakly.

U51¹(tt, n__length(x0), y1) → U52¹(isNat(length(x0)), activate(y1))
U51¹(tt, x0, y1) → U52¹(isNat(x0), activate(y1))
ISNATLIST(n__cons(V1, V2)) → U51¹(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U52¹(tt, n__isNatKind(x0)) → ISNATLIST(isNatKind(x0))
U51¹(tt, n__s(x0), y1) → U52¹(isNat(n__s(x0)), activate(y1))
U51¹(tt, n__0, y0) → U52¹(isNat(n__0), activate(y0))
U52¹(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
U51¹(tt, n__isNatIListKind(x0), y1) → U52¹(isNat(isNatIListKind(x0)), activate(y1))
U52¹(tt, x0) → ISNATLIST(x0)
U51¹(tt, n__isNatKind(x0), y1) → U52¹(isNat(isNatKind(x0)), activate(y1))
U51¹(tt, n__and(x0, x1), y1) → U52¹(isNat(and(x0, x1)), activate(y1))
U52¹(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
U52¹(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
U52¹(tt, n__length(x0)) → ISNATLIST(length(x0))
U52¹(tt, n__isNatIListKind(x0)) → ISNATLIST(isNatIListKind(x0))
U52¹(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))
U52¹(tt, n__and(x0, x1)) → ISNATLIST(and(x0, x1))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(ISNATLIST(x₁)) = x₁
POL(U11(x₁, x₂)) = 0
POL(U12(x₁)) = 0
POL(U21(x₁, x₂)) = 0
POL(U22(x₁)) = 0
POL(U31(x₁, x₂)) = 0
POL(U32(x₁)) = 0
POL(U41(x₁, x₂, x₃)) = x₂ + x₃
POL(U42(x₁, x₂)) = x₂
POL(U43(x₁)) = 0
POL(U51(x₁, x₂, x₃)) = 0
POL(U51¹(x₁, x₂, x₃)) = x₂ + x₃
POL(U52(x₁, x₂)) = 0
POL(U52¹(x₁, x₂)) = x₂
POL(U53(x₁)) = 0
POL(U61(x₁, x₂, x₃)) = 1
POL(U62(x₁, x₂)) = 1
POL(U63(x₁)) = 0
POL(U71(x₁, x₂)) = 0
POL(U81(x₁)) = 0
POL(U91(x₁, x₂, x₃, x₄)) = 1 + x₂ + x₄
POL(activate(x₁)) = x₁
POL(and(x₁, x₂)) = x₂
POL(cons(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = x₁
POL(isNatIList(x₁)) = x₁
POL(isNatIListKind(x₁)) = 0
POL(isNatKind(x₁)) = 0
POL(isNatList(x₁)) = 1
POL(length(x₁)) = x₁
POL(n__0) = 0
POL(n__and(x₁, x₂)) = x₂
POL(n__cons(x₁, x₂)) = x₁ + x₂
POL(n__isNatIListKind(x₁)) = 0
POL(n__isNatKind(x₁)) = 0
POL(n__length(x₁)) = x₁
POL(n__nil) = 0
POL(n__s(x₁)) = 0
POL(n__take(x₁, x₂)) = 1 + x₂
POL(n__zeros) = 0
POL(nil) = 0
POL(s(x₁)) = 0
POL(take(x₁, x₂)) = 1 + x₂
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

zeros → cons(0, n__zeros)
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U81(tt) → nil
U62(tt, V2) → U63(isNatIList(activate(V2)))
U61(tt, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U71(tt, L) → s(length(activate(L)))
U63(tt) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U43(tt) → tt
U53(tt) → tt
U52(tt, V2) → U53(isNatList(activate(V2)))
U32(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U42(tt, V2) → U43(isNatIList(activate(V2)))
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
isNatKind(n__0) → 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
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))
activate(n__0) → 0
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__zeros) → zeros
isNatKind(X) → n__isNatKind(X)
and(X1, X2) → n__and(X1, X2)
nil → n__nil
isNatIListKind(X) → n__isNatIListKind(X)
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(X) → n__length(X)
0 → n__0
take(X1, X2) → n__take(X1, X2)
zeros → n__zeros
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
activate(X) → X
activate(n__nil) → nil
activate(n__cons(X1, X2)) → cons(X1, X2)
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
activate(n__isNatIListKind(X)) → isNatIListKind(X)
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
and(tt, X) → activate(X)
activate(n__isNatKind(X)) → isNatKind(X)
isNatIListKind(n__take(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ AND

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ Narrowing

                                                                                                                            ↳ QDP

                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QDPOrderProof

                            ↳ QDP

                            ↳ QDP

                            ↳ QDP

                  ↳ QDP

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

U51¹(tt, n__length(x0), y1) → U52¹(isNat(length(x0)), activate(y1))
U51¹(tt, x0, y1) → U52¹(isNat(x0), activate(y1))
ISNATLIST(n__cons(V1, V2)) → U51¹(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U52¹(tt, n__isNatKind(x0)) → ISNATLIST(isNatKind(x0))
U51¹(tt, n__s(x0), y1) → U52¹(isNat(n__s(x0)), activate(y1))
U51¹(tt, n__0, y0) → U52¹(isNat(n__0), activate(y0))
U52¹(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
U51¹(tt, n__isNatIListKind(x0), y1) → U52¹(isNat(isNatIListKind(x0)), activate(y1))
U52¹(tt, x0) → ISNATLIST(x0)
U51¹(tt, n__isNatKind(x0), y1) → U52¹(isNat(isNatKind(x0)), activate(y1))
U51¹(tt, n__and(x0, x1), y1) → U52¹(isNat(and(x0, x1)), activate(y1))
U52¹(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
U52¹(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
U52¹(tt, n__length(x0)) → ISNATLIST(length(x0))
U52¹(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))
U52¹(tt, n__isNatIListKind(x0)) → ISNATLIST(isNatIListKind(x0))
U52¹(tt, n__and(x0, x1)) → ISNATLIST(and(x0, x1))

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, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
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)))
isNatIListKind(n__take(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))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

U51¹(tt, n__length(x0), y1) → U52¹(isNat(length(x0)), activate(y1))

The remaining pairs can at least be oriented weakly.

U51¹(tt, x0, y1) → U52¹(isNat(x0), activate(y1))
ISNATLIST(n__cons(V1, V2)) → U51¹(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U52¹(tt, n__isNatKind(x0)) → ISNATLIST(isNatKind(x0))
U51¹(tt, n__s(x0), y1) → U52¹(isNat(n__s(x0)), activate(y1))
U51¹(tt, n__0, y0) → U52¹(isNat(n__0), activate(y0))
U52¹(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
U51¹(tt, n__isNatIListKind(x0), y1) → U52¹(isNat(isNatIListKind(x0)), activate(y1))
U52¹(tt, x0) → ISNATLIST(x0)
U51¹(tt, n__isNatKind(x0), y1) → U52¹(isNat(isNatKind(x0)), activate(y1))
U51¹(tt, n__and(x0, x1), y1) → U52¹(isNat(and(x0, x1)), activate(y1))
U52¹(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
U52¹(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
U52¹(tt, n__length(x0)) → ISNATLIST(length(x0))
U52¹(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))
U52¹(tt, n__isNatIListKind(x0)) → ISNATLIST(isNatIListKind(x0))
U52¹(tt, n__and(x0, x1)) → ISNATLIST(and(x0, x1))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(ISNATLIST(x₁)) = x₁
POL(U11(x₁, x₂)) = 0
POL(U12(x₁)) = 0
POL(U21(x₁, x₂)) = 0
POL(U22(x₁)) = 0
POL(U31(x₁, x₂)) = x₂
POL(U32(x₁)) = x₁
POL(U41(x₁, x₂, x₃)) = 0
POL(U42(x₁, x₂)) = 0
POL(U43(x₁)) = 0
POL(U51(x₁, x₂, x₃)) = x₂ + x₃
POL(U51¹(x₁, x₂, x₃)) = x₂ + x₃
POL(U52(x₁, x₂)) = x₂
POL(U52¹(x₁, x₂)) = x₂
POL(U53(x₁)) = 0
POL(U61(x₁, x₂, x₃)) = 0
POL(U62(x₁, x₂)) = 0
POL(U63(x₁)) = 0
POL(U71(x₁, x₂)) = 0
POL(U81(x₁)) = 0
POL(U91(x₁, x₂, x₃, x₄)) = x₂ + x₄
POL(activate(x₁)) = x₁
POL(and(x₁, x₂)) = x₂
POL(cons(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = x₁
POL(isNatIListKind(x₁)) = 0
POL(isNatKind(x₁)) = 0
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = 1
POL(n__0) = 0
POL(n__and(x₁, x₂)) = x₂
POL(n__cons(x₁, x₂)) = x₁ + x₂
POL(n__isNatIListKind(x₁)) = 0
POL(n__isNatKind(x₁)) = 0
POL(n__length(x₁)) = 1
POL(n__nil) = 0
POL(n__s(x₁)) = 0
POL(n__take(x₁, x₂)) = x₂
POL(n__zeros) = 0
POL(nil) = 0
POL(s(x₁)) = 0
POL(take(x₁, x₂)) = x₂
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

zeros → cons(0, n__zeros)
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U81(tt) → nil
U62(tt, V2) → U63(isNatIList(activate(V2)))
U61(tt, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U71(tt, L) → s(length(activate(L)))
U63(tt) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U43(tt) → tt
U53(tt) → tt
U52(tt, V2) → U53(isNatList(activate(V2)))
U32(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U42(tt, V2) → U43(isNatIList(activate(V2)))
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
isNatKind(n__0) → 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
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))
activate(n__0) → 0
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__zeros) → zeros
isNatKind(X) → n__isNatKind(X)
and(X1, X2) → n__and(X1, X2)
nil → n__nil
isNatIListKind(X) → n__isNatIListKind(X)
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(X) → n__length(X)
0 → n__0
take(X1, X2) → n__take(X1, X2)
zeros → n__zeros
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
activate(X) → X
activate(n__nil) → nil
activate(n__cons(X1, X2)) → cons(X1, X2)
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
activate(n__isNatIListKind(X)) → isNatIListKind(X)
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
and(tt, X) → activate(X)
activate(n__isNatKind(X)) → isNatKind(X)
isNatIListKind(n__take(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ AND

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ Narrowing

                                                                                                                            ↳ QDP

                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QDPOrderProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ QDPOrderProof

                            ↳ QDP

                            ↳ QDP

                            ↳ QDP

                  ↳ QDP

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

U51¹(tt, x0, y1) → U52¹(isNat(x0), activate(y1))
ISNATLIST(n__cons(V1, V2)) → U51¹(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U52¹(tt, n__isNatKind(x0)) → ISNATLIST(isNatKind(x0))
U51¹(tt, n__s(x0), y1) → U52¹(isNat(n__s(x0)), activate(y1))
U51¹(tt, n__0, y0) → U52¹(isNat(n__0), activate(y0))
U52¹(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
U51¹(tt, n__isNatIListKind(x0), y1) → U52¹(isNat(isNatIListKind(x0)), activate(y1))
U52¹(tt, x0) → ISNATLIST(x0)
U51¹(tt, n__isNatKind(x0), y1) → U52¹(isNat(isNatKind(x0)), activate(y1))
U51¹(tt, n__and(x0, x1), y1) → U52¹(isNat(and(x0, x1)), activate(y1))
U52¹(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
U52¹(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
U52¹(tt, n__length(x0)) → ISNATLIST(length(x0))
U52¹(tt, n__isNatIListKind(x0)) → ISNATLIST(isNatIListKind(x0))
U52¹(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))
U52¹(tt, n__and(x0, x1)) → ISNATLIST(and(x0, x1))

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, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
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)))
isNatIListKind(n__take(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))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

U51¹(tt, n__s(x0), y1) → U52¹(isNat(n__s(x0)), activate(y1))

The remaining pairs can at least be oriented weakly.

U51¹(tt, x0, y1) → U52¹(isNat(x0), activate(y1))
ISNATLIST(n__cons(V1, V2)) → U51¹(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U52¹(tt, n__isNatKind(x0)) → ISNATLIST(isNatKind(x0))
U51¹(tt, n__0, y0) → U52¹(isNat(n__0), activate(y0))
U52¹(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
U51¹(tt, n__isNatIListKind(x0), y1) → U52¹(isNat(isNatIListKind(x0)), activate(y1))
U52¹(tt, x0) → ISNATLIST(x0)
U51¹(tt, n__isNatKind(x0), y1) → U52¹(isNat(isNatKind(x0)), activate(y1))
U51¹(tt, n__and(x0, x1), y1) → U52¹(isNat(and(x0, x1)), activate(y1))
U52¹(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
U52¹(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
U52¹(tt, n__length(x0)) → ISNATLIST(length(x0))
U52¹(tt, n__isNatIListKind(x0)) → ISNATLIST(isNatIListKind(x0))
U52¹(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))
U52¹(tt, n__and(x0, x1)) → ISNATLIST(and(x0, x1))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(ISNATLIST(x₁)) = x₁
POL(U11(x₁, x₂)) = 0
POL(U12(x₁)) = 0
POL(U21(x₁, x₂)) = 0
POL(U22(x₁)) = 0
POL(U31(x₁, x₂)) = 0
POL(U32(x₁)) = 0
POL(U41(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(U42(x₁, x₂)) = 0
POL(U43(x₁)) = 0
POL(U51(x₁, x₂, x₃)) = 1
POL(U51¹(x₁, x₂, x₃)) = x₂ + x₃
POL(U52(x₁, x₂)) = 0
POL(U52¹(x₁, x₂)) = x₂
POL(U53(x₁)) = 0
POL(U61(x₁, x₂, x₃)) = 1
POL(U62(x₁, x₂)) = 1
POL(U63(x₁)) = 1
POL(U71(x₁, x₂)) = 1
POL(U81(x₁)) = 0
POL(U91(x₁, x₂, x₃, x₄)) = x₂ + x₄
POL(activate(x₁)) = x₁
POL(and(x₁, x₂)) = x₂
POL(cons(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 1 + x₁
POL(isNatIListKind(x₁)) = 0
POL(isNatKind(x₁)) = 0
POL(isNatList(x₁)) = 1
POL(length(x₁)) = 1 + x₁
POL(n__0) = 0
POL(n__and(x₁, x₂)) = x₂
POL(n__cons(x₁, x₂)) = x₁ + x₂
POL(n__isNatIListKind(x₁)) = 0
POL(n__isNatKind(x₁)) = 0
POL(n__length(x₁)) = 1 + x₁
POL(n__nil) = 0
POL(n__s(x₁)) = 1
POL(n__take(x₁, x₂)) = x₂
POL(n__zeros) = 0
POL(nil) = 0
POL(s(x₁)) = 1
POL(take(x₁, x₂)) = x₂
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

zeros → cons(0, n__zeros)
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U81(tt) → nil
U62(tt, V2) → U63(isNatIList(activate(V2)))
U61(tt, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U71(tt, L) → s(length(activate(L)))
U63(tt) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U43(tt) → tt
U53(tt) → tt
U52(tt, V2) → U53(isNatList(activate(V2)))
U32(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U42(tt, V2) → U43(isNatIList(activate(V2)))
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
isNatKind(n__0) → 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
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))
activate(n__0) → 0
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__zeros) → zeros
isNatKind(X) → n__isNatKind(X)
and(X1, X2) → n__and(X1, X2)
nil → n__nil
isNatIListKind(X) → n__isNatIListKind(X)
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(X) → n__length(X)
0 → n__0
take(X1, X2) → n__take(X1, X2)
zeros → n__zeros
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
activate(X) → X
activate(n__nil) → nil
activate(n__cons(X1, X2)) → cons(X1, X2)
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
activate(n__isNatIListKind(X)) → isNatIListKind(X)
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
and(tt, X) → activate(X)
activate(n__isNatKind(X)) → isNatKind(X)
isNatIListKind(n__take(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ AND

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ Narrowing

                                                                                                                            ↳ QDP

                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QDPOrderProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ QDPOrderProof

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ QDPOrderProof

                            ↳ QDP

                            ↳ QDP

                            ↳ QDP

                  ↳ QDP

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

U51¹(tt, x0, y1) → U52¹(isNat(x0), activate(y1))
ISNATLIST(n__cons(V1, V2)) → U51¹(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U52¹(tt, n__isNatKind(x0)) → ISNATLIST(isNatKind(x0))
U51¹(tt, n__0, y0) → U52¹(isNat(n__0), activate(y0))
U52¹(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
U51¹(tt, n__isNatIListKind(x0), y1) → U52¹(isNat(isNatIListKind(x0)), activate(y1))
U52¹(tt, x0) → ISNATLIST(x0)
U51¹(tt, n__isNatKind(x0), y1) → U52¹(isNat(isNatKind(x0)), activate(y1))
U51¹(tt, n__and(x0, x1), y1) → U52¹(isNat(and(x0, x1)), activate(y1))
U52¹(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
U52¹(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
U52¹(tt, n__length(x0)) → ISNATLIST(length(x0))
U52¹(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))
U52¹(tt, n__isNatIListKind(x0)) → ISNATLIST(isNatIListKind(x0))
U52¹(tt, n__and(x0, x1)) → ISNATLIST(and(x0, x1))

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, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
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)))
isNatIListKind(n__take(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))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

U51¹(tt, n__isNatIListKind(x0), y1) → U52¹(isNat(isNatIListKind(x0)), activate(y1))
U51¹(tt, n__isNatKind(x0), y1) → U52¹(isNat(isNatKind(x0)), activate(y1))

The remaining pairs can at least be oriented weakly.

U51¹(tt, x0, y1) → U52¹(isNat(x0), activate(y1))
ISNATLIST(n__cons(V1, V2)) → U51¹(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U52¹(tt, n__isNatKind(x0)) → ISNATLIST(isNatKind(x0))
U51¹(tt, n__0, y0) → U52¹(isNat(n__0), activate(y0))
U52¹(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
U52¹(tt, x0) → ISNATLIST(x0)
U51¹(tt, n__and(x0, x1), y1) → U52¹(isNat(and(x0, x1)), activate(y1))
U52¹(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
U52¹(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
U52¹(tt, n__length(x0)) → ISNATLIST(length(x0))
U52¹(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))
U52¹(tt, n__isNatIListKind(x0)) → ISNATLIST(isNatIListKind(x0))
U52¹(tt, n__and(x0, x1)) → ISNATLIST(and(x0, x1))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(ISNATLIST(x₁)) = x₁
POL(U11(x₁, x₂)) = 0
POL(U12(x₁)) = 0
POL(U21(x₁, x₂)) = 0
POL(U22(x₁)) = 0
POL(U31(x₁, x₂)) = 1 + x₂
POL(U32(x₁)) = x₁
POL(U41(x₁, x₂, x₃)) = x₁
POL(U42(x₁, x₂)) = 0
POL(U43(x₁)) = 0
POL(U51(x₁, x₂, x₃)) = x₂
POL(U51¹(x₁, x₂, x₃)) = x₂ + x₃
POL(U52(x₁, x₂)) = 0
POL(U52¹(x₁, x₂)) = x₂
POL(U53(x₁)) = 0
POL(U61(x₁, x₂, x₃)) = 0
POL(U62(x₁, x₂)) = 0
POL(U63(x₁)) = 0
POL(U71(x₁, x₂)) = x₂
POL(U81(x₁)) = 0
POL(U91(x₁, x₂, x₃, x₄)) = x₂ + x₄
POL(activate(x₁)) = x₁
POL(and(x₁, x₂)) = x₂
POL(cons(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 1 + x₁
POL(isNatIListKind(x₁)) = 1
POL(isNatKind(x₁)) = 1
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = x₁
POL(n__0) = 0
POL(n__and(x₁, x₂)) = x₂
POL(n__cons(x₁, x₂)) = x₁ + x₂
POL(n__isNatIListKind(x₁)) = 1
POL(n__isNatKind(x₁)) = 1
POL(n__length(x₁)) = x₁
POL(n__nil) = 0
POL(n__s(x₁)) = 0
POL(n__take(x₁, x₂)) = x₂
POL(n__zeros) = 0
POL(nil) = 0
POL(s(x₁)) = 0
POL(take(x₁, x₂)) = x₂
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

zeros → cons(0, n__zeros)
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U81(tt) → nil
U62(tt, V2) → U63(isNatIList(activate(V2)))
U61(tt, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U71(tt, L) → s(length(activate(L)))
U63(tt) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U43(tt) → tt
U53(tt) → tt
U52(tt, V2) → U53(isNatList(activate(V2)))
U32(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U42(tt, V2) → U43(isNatIList(activate(V2)))
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
isNatKind(n__0) → 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
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))
activate(n__0) → 0
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__zeros) → zeros
isNatKind(X) → n__isNatKind(X)
and(X1, X2) → n__and(X1, X2)
nil → n__nil
isNatIListKind(X) → n__isNatIListKind(X)
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(X) → n__length(X)
0 → n__0
take(X1, X2) → n__take(X1, X2)
zeros → n__zeros
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
activate(X) → X
activate(n__nil) → nil
activate(n__cons(X1, X2)) → cons(X1, X2)
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
activate(n__isNatIListKind(X)) → isNatIListKind(X)
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
and(tt, X) → activate(X)
activate(n__isNatKind(X)) → isNatKind(X)
isNatIListKind(n__take(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ AND

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ Narrowing

                                                                                                                            ↳ QDP

                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QDPOrderProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ QDPOrderProof

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ QDPOrderProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ QDPOrderProof

                            ↳ QDP

                            ↳ QDP

                            ↳ QDP

                  ↳ QDP

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

U51¹(tt, x0, y1) → U52¹(isNat(x0), activate(y1))
ISNATLIST(n__cons(V1, V2)) → U51¹(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U52¹(tt, n__isNatKind(x0)) → ISNATLIST(isNatKind(x0))
U51¹(tt, n__0, y0) → U52¹(isNat(n__0), activate(y0))
U52¹(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
U52¹(tt, x0) → ISNATLIST(x0)
U51¹(tt, n__and(x0, x1), y1) → U52¹(isNat(and(x0, x1)), activate(y1))
U52¹(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
U52¹(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
U52¹(tt, n__length(x0)) → ISNATLIST(length(x0))
U52¹(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))
U52¹(tt, n__isNatIListKind(x0)) → ISNATLIST(isNatIListKind(x0))
U52¹(tt, n__and(x0, x1)) → ISNATLIST(and(x0, x1))

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, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
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)))
isNatIListKind(n__take(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))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

U52¹(tt, n__length(x0)) → ISNATLIST(length(x0))

The remaining pairs can at least be oriented weakly.

U51¹(tt, x0, y1) → U52¹(isNat(x0), activate(y1))
ISNATLIST(n__cons(V1, V2)) → U51¹(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U52¹(tt, n__isNatKind(x0)) → ISNATLIST(isNatKind(x0))
U51¹(tt, n__0, y0) → U52¹(isNat(n__0), activate(y0))
U52¹(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
U52¹(tt, x0) → ISNATLIST(x0)
U51¹(tt, n__and(x0, x1), y1) → U52¹(isNat(and(x0, x1)), activate(y1))
U52¹(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
U52¹(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
U52¹(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))
U52¹(tt, n__isNatIListKind(x0)) → ISNATLIST(isNatIListKind(x0))
U52¹(tt, n__and(x0, x1)) → ISNATLIST(and(x0, x1))

Used ordering: Matrix interpretation [3]:
Non-tuple symbols:

M( U62(x₁, x₂) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( U52(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	1	/

x₂

M( U61(x₁, ..., x₃) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

/	0	0	\
\	0	0	/

x₃

M( U81(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U41(x₁, ..., x₃) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

/	0	0	\
\	0	0	/

x₃

M( activate(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( n__nil ) =

/	0	\
\	0	/

M( and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	0	1	/

x₂

M( n__s(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( take(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	1	/

x₂

M( U71(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( U21(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( isNatIListKind(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( tt ) =

/	0	\
\	0	/

M( isNatList(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( zeros ) =

/	0	\
\	0	/

M( isNatIList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U12(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( s(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U51(x₁, ..., x₃) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

/	0	0	\
\	0	1	/

x₃

M( nil ) =

/	0	\
\	0	/

M( U91(x₁, ..., x₄) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	1	/

x₂

/	0	0	\
\	0	0	/

x₃

/	0	0	\
\	0	0	/

x₄

M( U31(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( n__length(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U42(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( n__isNatKind(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( n__zeros ) =

/	0	\
\	0	/

M( n__cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	1	1	/

x₂

M( U11(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( isNatKind(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U63(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( 0 ) =

/	0	\
\	0	/

M( U43(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( n__and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	0	1	/

x₂

M( cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	1	1	/

x₂

M( U22(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U53(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( n__0 ) =

/	0	\
\	0	/

M( U32(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( n__take(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	1	/

x₂

M( n__isNatIListKind(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( length(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

Tuple symbols:

M( U51¹(x₁, ..., x₃) ) =

[

]

x₁

[

]

x₂

[

]

x₃

M( U52¹(x₁, x₂) ) =

[

]

x₁

[

]

x₂

M( ISNATLIST(x₁) ) =

[

]

x₁

Matrix type:
We used a basic matrix type which is not further parametrizeable.

As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
The following usable rules [17] were oriented:

zeros → cons(0, n__zeros)
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U81(tt) → nil
U62(tt, V2) → U63(isNatIList(activate(V2)))
U61(tt, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U71(tt, L) → s(length(activate(L)))
U63(tt) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U43(tt) → tt
U53(tt) → tt
U52(tt, V2) → U53(isNatList(activate(V2)))
U32(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U42(tt, V2) → U43(isNatIList(activate(V2)))
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
isNatKind(n__0) → 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
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))
activate(n__0) → 0
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__zeros) → zeros
isNatKind(X) → n__isNatKind(X)
and(X1, X2) → n__and(X1, X2)
nil → n__nil
isNatIListKind(X) → n__isNatIListKind(X)
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(X) → n__length(X)
0 → n__0
take(X1, X2) → n__take(X1, X2)
zeros → n__zeros
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
activate(X) → X
activate(n__nil) → nil
activate(n__cons(X1, X2)) → cons(X1, X2)
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
activate(n__isNatIListKind(X)) → isNatIListKind(X)
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
and(tt, X) → activate(X)
activate(n__isNatKind(X)) → isNatKind(X)
isNatIListKind(n__take(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ AND

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ Narrowing

                                                                                                                            ↳ QDP

                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QDPOrderProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ QDPOrderProof

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ QDPOrderProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ QDPOrderProof

                            ↳ QDP

                            ↳ QDP

                            ↳ QDP

                  ↳ QDP

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

U52¹(tt, x0) → ISNATLIST(x0)
U51¹(tt, n__and(x0, x1), y1) → U52¹(isNat(and(x0, x1)), activate(y1))
U52¹(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
U51¹(tt, x0, y1) → U52¹(isNat(x0), activate(y1))
ISNATLIST(n__cons(V1, V2)) → U51¹(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U52¹(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
U52¹(tt, n__isNatKind(x0)) → ISNATLIST(isNatKind(x0))
U52¹(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
U51¹(tt, n__0, y0) → U52¹(isNat(n__0), activate(y0))
U52¹(tt, n__isNatIListKind(x0)) → ISNATLIST(isNatIListKind(x0))
U52¹(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))
U52¹(tt, n__and(x0, x1)) → ISNATLIST(and(x0, x1))

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, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
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)))
isNatIListKind(n__take(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))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

U52¹(tt, n__isNatIListKind(x0)) → ISNATLIST(isNatIListKind(x0))

The remaining pairs can at least be oriented weakly.

U52¹(tt, x0) → ISNATLIST(x0)
U51¹(tt, n__and(x0, x1), y1) → U52¹(isNat(and(x0, x1)), activate(y1))
U52¹(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
U51¹(tt, x0, y1) → U52¹(isNat(x0), activate(y1))
ISNATLIST(n__cons(V1, V2)) → U51¹(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U52¹(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
U52¹(tt, n__isNatKind(x0)) → ISNATLIST(isNatKind(x0))
U52¹(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
U51¹(tt, n__0, y0) → U52¹(isNat(n__0), activate(y0))
U52¹(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))
U52¹(tt, n__and(x0, x1)) → ISNATLIST(and(x0, x1))

Used ordering: Matrix interpretation [3]:
Non-tuple symbols:

M( U62(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( U52(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( U61(x₁, ..., x₃) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

/	0	0	\
\	0	0	/

x₃

M( U81(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U41(x₁, ..., x₃) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

/	0	0	\
\	0	0	/

x₃

M( activate(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( n__nil ) =

/	0	\
\	0	/

M( and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	0	1	/

x₂

M( n__s(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

M( take(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( U71(x₁, x₂) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	1	\
\	0	0	/

x₂

M( U21(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( isNatIListKind(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( tt ) =

/	0	\
\	0	/

M( isNatList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( zeros ) =

/	0	\
\	0	/

M( isNatIList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U12(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( s(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

M( isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U51(x₁, ..., x₃) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

/	0	0	\
\	0	0	/

x₃

M( nil ) =

/	0	\
\	0	/

M( U91(x₁, ..., x₄) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

/	0	0	\
\	0	0	/

x₃

/	0	0	\
\	0	0	/

x₄

M( U31(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( n__length(x₁) ) =

/	1	\
\	0	/

/	0	1	\
\	0	0	/

x₁

M( U42(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( n__isNatKind(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

M( n__zeros ) =

/	0	\
\	0	/

M( n__cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	1	1	/

x₂

M( U11(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( isNatKind(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

M( U63(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( 0 ) =

/	0	\
\	0	/

M( U43(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( n__and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	0	1	/

x₂

M( cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	1	1	/

x₂

M( U22(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U53(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( n__0 ) =

/	0	\
\	0	/

M( U32(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( n__take(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( n__isNatIListKind(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( length(x₁) ) =

/	1	\
\	0	/

/	0	1	\
\	0	0	/

x₁

Tuple symbols:

M( U51¹(x₁, ..., x₃) ) =

[

]

x₁

[

]

x₂

[

]

x₃

M( U52¹(x₁, x₂) ) =

[

]

x₁

[

]

x₂

M( ISNATLIST(x₁) ) =

[

]

x₁

zeros → cons(0, n__zeros)
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U81(tt) → nil
U62(tt, V2) → U63(isNatIList(activate(V2)))
U61(tt, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U71(tt, L) → s(length(activate(L)))
U63(tt) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U43(tt) → tt
U53(tt) → tt
U52(tt, V2) → U53(isNatList(activate(V2)))
U32(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U42(tt, V2) → U43(isNatIList(activate(V2)))
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
isNatKind(n__0) → 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
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))
activate(n__0) → 0
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__zeros) → zeros
isNatKind(X) → n__isNatKind(X)
and(X1, X2) → n__and(X1, X2)
nil → n__nil
isNatIListKind(X) → n__isNatIListKind(X)
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(X) → n__length(X)
0 → n__0
take(X1, X2) → n__take(X1, X2)
zeros → n__zeros
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
activate(X) → X
activate(n__nil) → nil
activate(n__cons(X1, X2)) → cons(X1, X2)
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
activate(n__isNatIListKind(X)) → isNatIListKind(X)
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
and(tt, X) → activate(X)
activate(n__isNatKind(X)) → isNatKind(X)
isNatIListKind(n__take(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ AND

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ Narrowing

                                                                                                                            ↳ QDP

                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QDPOrderProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ QDPOrderProof

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ QDPOrderProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ QDPOrderProof

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ QDPOrderProof

                            ↳ QDP

                            ↳ QDP

                            ↳ QDP

                  ↳ QDP

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

U52¹(tt, x0) → ISNATLIST(x0)
U51¹(tt, n__and(x0, x1), y1) → U52¹(isNat(and(x0, x1)), activate(y1))
U51¹(tt, x0, y1) → U52¹(isNat(x0), activate(y1))
U52¹(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
ISNATLIST(n__cons(V1, V2)) → U51¹(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U52¹(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
U52¹(tt, n__isNatKind(x0)) → ISNATLIST(isNatKind(x0))
U51¹(tt, n__0, y0) → U52¹(isNat(n__0), activate(y0))
U52¹(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
U52¹(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))
U52¹(tt, n__and(x0, x1)) → ISNATLIST(and(x0, x1))

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, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
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)))
isNatIListKind(n__take(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))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

U52¹(tt, n__isNatKind(x0)) → ISNATLIST(isNatKind(x0))

The remaining pairs can at least be oriented weakly.

U52¹(tt, x0) → ISNATLIST(x0)
U51¹(tt, n__and(x0, x1), y1) → U52¹(isNat(and(x0, x1)), activate(y1))
U51¹(tt, x0, y1) → U52¹(isNat(x0), activate(y1))
U52¹(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
ISNATLIST(n__cons(V1, V2)) → U51¹(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U52¹(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
U51¹(tt, n__0, y0) → U52¹(isNat(n__0), activate(y0))
U52¹(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
U52¹(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))
U52¹(tt, n__and(x0, x1)) → ISNATLIST(and(x0, x1))

Used ordering: Matrix interpretation [3]:
Non-tuple symbols:

M( U62(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( U52(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( U61(x₁, ..., x₃) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

/	0	0	\
\	0	0	/

x₃

M( U81(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U41(x₁, ..., x₃) ) =

/	1	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

/	1	1	\
\	1	1	/

x₃

M( activate(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( n__nil ) =

/	0	\
\	0	/

M( and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	0	1	/

x₂

M( n__s(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( take(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

/	0	0	\
\	0	0	/

x₂

M( U71(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	1	1	/

x₂

M( U21(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( isNatIListKind(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( tt ) =

/	0	\
\	0	/

M( isNatList(x₁) ) =

/	0	\
\	0	/

/	0	1	\
\	0	0	/

x₁

M( zeros ) =

/	0	\
\	0	/

M( isNatIList(x₁) ) =

/	1	\
\	1	/

/	0	1	\
\	0	1	/

x₁

M( U12(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( s(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U51(x₁, ..., x₃) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

/	0	1	\
\	0	0	/

x₃

M( nil ) =

/	0	\
\	0	/

M( U91(x₁, ..., x₄) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

/	0	0	\
\	0	1	/

x₃

/	0	0	\
\	0	0	/

x₄

M( U31(x₁, x₂) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( n__length(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	1	1	/

x₁

M( U42(x₁, x₂) ) =

/	1	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	1	/

x₂

M( n__isNatKind(x₁) ) =

/	1	\
\	1	/

/	0	0	\
\	0	0	/

x₁

M( n__zeros ) =

/	0	\
\	0	/

M( n__cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	1	1	/

x₂

M( U11(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( isNatKind(x₁) ) =

/	1	\
\	1	/

/	0	0	\
\	0	0	/

x₁

M( U63(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( 0 ) =

/	0	\
\	0	/

M( U43(x₁) ) =

/	1	\
\	1	/

/	0	0	\
\	0	0	/

x₁

M( n__and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	0	1	/

x₂

M( cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	1	1	/

x₂

M( U22(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U53(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( n__0 ) =

/	0	\
\	0	/

M( U32(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( n__take(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

/	0	0	\
\	0	0	/

x₂

M( n__isNatIListKind(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( length(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	1	1	/

x₁

Tuple symbols:

M( U51¹(x₁, ..., x₃) ) =

[

]

x₁

[

]

x₂

[

]

x₃

M( U52¹(x₁, x₂) ) =

[

]

x₁

[

]

x₂

M( ISNATLIST(x₁) ) =

[

]

x₁

zeros → cons(0, n__zeros)
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
U81(tt) → nil
U62(tt, V2) → U63(isNatIList(activate(V2)))
U61(tt, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U71(tt, L) → s(length(activate(L)))
U63(tt) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U43(tt) → tt
U53(tt) → tt
U52(tt, V2) → U53(isNatList(activate(V2)))
U32(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U42(tt, V2) → U43(isNatIList(activate(V2)))
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
isNatKind(n__0) → 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
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))
activate(n__0) → 0
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__zeros) → zeros
isNatKind(X) → n__isNatKind(X)
and(X1, X2) → n__and(X1, X2)
nil → n__nil
isNatIListKind(X) → n__isNatIListKind(X)
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(X) → n__length(X)
0 → n__0
take(X1, X2) → n__take(X1, X2)
zeros → n__zeros
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
activate(X) → X
activate(n__nil) → nil
activate(n__cons(X1, X2)) → cons(X1, X2)
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
activate(n__isNatIListKind(X)) → isNatIListKind(X)
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
and(tt, X) → activate(X)
activate(n__isNatKind(X)) → isNatKind(X)
isNatIListKind(n__take(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ AND

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ Narrowing

                                                                                                                            ↳ QDP

                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ QDPOrderProof

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ QDPOrderProof

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ QDPOrderProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ QDPOrderProof

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ QDPOrderProof

                                                                                                                                                            ↳ QDP

                            ↳ QDP

                            ↳ QDP

                            ↳ QDP

                  ↳ QDP

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

U52¹(tt, x0) → ISNATLIST(x0)
U51¹(tt, n__and(x0, x1), y1) → U52¹(isNat(and(x0, x1)), activate(y1))
U52¹(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
U51¹(tt, x0, y1) → U52¹(isNat(x0), activate(y1))
ISNATLIST(n__cons(V1, V2)) → U51¹(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U52¹(tt, n__take(x0, x1)) → ISNATLIST(take(x0, x1))
U52¹(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
U51¹(tt, n__0, y0) → U52¹(isNat(n__0), activate(y0))
U52¹(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))
U52¹(tt, n__and(x0, x1)) → ISNATLIST(and(x0, x1))

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, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
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)))
isNatIListKind(n__take(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))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ AND

                            ↳ QDP

                            ↳ QDP

                            ↳ QDP

                            ↳ QDP

                  ↳ QDP

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

U42¹(tt, V2) → ISNATILIST(activate(V2))
ISNATILIST(n__cons(V1, V2)) → U41¹(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U41¹(tt, V1, V2) → U42¹(isNat(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, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
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)))
isNatIListKind(n__take(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))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ AND

                            ↳ QDP

                            ↳ QDP

                            ↳ QDP

                            ↳ QDP

                  ↳ QDP

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

ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATILISTKIND(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ISNATILISTKIND(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
AND(tt, X) → ACTIVATE(X)
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)

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, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
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)))
isNatIListKind(n__take(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))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ AND

                            ↳ QDP

                            ↳ QDP

                            ↳ QDP

                            ↳ QDP

                  ↳ QDP

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

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

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, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
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)))
isNatIListKind(n__take(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))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

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

LENGTH(cons(N, L)) → U71¹(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
U71¹(tt, L) → LENGTH(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, V1, V2) → U62(isNat(activate(V1)), activate(V2))
U62(tt, V2) → U63(isNatIList(activate(V2)))
U63(tt) → tt
U71(tt, L) → s(length(activate(L)))
U81(tt) → nil
U91(tt, IL, M, N) → cons(activate(N), n__take(activate(M), activate(IL)))
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)))
isNatIListKind(n__take(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))
isNatList(n__take(V1, V2)) → U61(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U71(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
take(0, IL) → U81(and(isNatIList(IL), n__isNatIListKind(IL)))
take(s(M), cons(N, IL)) → U91(and(and(isNatIList(activate(IL)), n__isNatIListKind(activate(IL))), n__and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N)))), activate(IL), M, N)
zeros → n__zeros
take(X1, X2) → n__take(X1, X2)
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X

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