Termination w.r.t. Q of the following Term Rewriting System could be disproven:

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

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

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

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

Q is empty.

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

U921(tt, L, N) → ACTIVATE(L)
ISNATILIST(V) → ACTIVATE(V)
U811(tt, V1, V2) → ISNATKIND(activate(V1))
ACTIVATE(n__0) → 01
U941(tt, L) → S(length(activate(L)))
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__length(V1)) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
U511(tt, V2) → ISNATILISTKIND(activate(V2))
U931(tt, L, N) → ACTIVATE(N)
ISNAT(n__length(V1)) → U111(isNatIListKind(activate(V1)), activate(V1))
ISNATKIND(n__length(V1)) → ISNATILISTKIND(activate(V1))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__s(X)) → S(activate(X))
ACTIVATE(n__cons(X1, X2)) → CONS(activate(X1), X2)
U431(tt, V1, V2) → ACTIVATE(V2)
U121(tt, V1) → U131(isNatList(activate(V1)))
U311(tt, V) → U321(isNatIListKind(activate(V)), activate(V))
U431(tt, V1, V2) → ACTIVATE(V1)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
U221(tt, V1) → U231(isNat(activate(V1)))
U451(tt, V2) → U461(isNatIList(activate(V2)))
U121(tt, V1) → ISNATLIST(activate(V1))
U831(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U911(tt, L, N) → ACTIVATE(N)
U911(tt, L, N) → ACTIVATE(L)
U921(tt, L, N) → U931(isNat(activate(N)), activate(L), activate(N))
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
U431(tt, V1, V2) → ISNATILISTKIND(activate(V2))
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V1)
U821(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U111(tt, V1) → ACTIVATE(V1)
U321(tt, V) → ISNATLIST(activate(V))
U921(tt, L, N) → ACTIVATE(N)
U911(tt, L, N) → U921(isNatIListKind(activate(L)), activate(L), activate(N))
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V2)
U821(tt, V1, V2) → U831(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U821(tt, V1, V2) → ACTIVATE(V2)
U411(tt, V1, V2) → ACTIVATE(V1)
U221(tt, V1) → ISNAT(activate(V1))
ISNATLIST(n__cons(V1, V2)) → U811(isNatKind(activate(V1)), activate(V1), activate(V2))
U221(tt, V1) → ACTIVATE(V1)
U841(tt, V1, V2) → U851(isNat(activate(V1)), activate(V2))
U841(tt, V1, V2) → ISNAT(activate(V1))
U311(tt, V) → ISNATILISTKIND(activate(V))
U511(tt, V2) → U521(isNatIListKind(activate(V2)))
U211(tt, V1) → ACTIVATE(V1)
U321(tt, V) → ACTIVATE(V)
ZEROS → CONS(0, n__zeros)
U421(tt, V1, V2) → ACTIVATE(V2)
U441(tt, V1, V2) → ACTIVATE(V1)
U431(tt, V1, V2) → U441(isNatIListKind(activate(V2)), activate(V1), activate(V2))
ISNATKIND(n__length(V1)) → U611(isNatIListKind(activate(V1)))
U211(tt, V1) → ISNATKIND(activate(V1))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
U451(tt, V2) → ACTIVATE(V2)
ACTIVATE(n__s(X)) → ACTIVATE(X)
U411(tt, V1, V2) → ISNATKIND(activate(V1))
U451(tt, V2) → ISNATILIST(activate(V2))
U821(tt, V1, V2) → ACTIVATE(V1)
U411(tt, V1, V2) → U421(isNatKind(activate(V1)), activate(V1), activate(V2))
U421(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U831(tt, V1, V2) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILISTKIND(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATLIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ACTIVATE(n__zeros) → ZEROS
LENGTH(nil) → 01
U931(tt, L, N) → ISNATKIND(activate(N))
U811(tt, V1, V2) → ACTIVATE(V2)
U921(tt, L, N) → ISNAT(activate(N))
U441(tt, V1, V2) → U451(isNat(activate(V1)), activate(V2))
U931(tt, L, N) → ACTIVATE(L)
ACTIVATE(n__nil) → NIL
U421(tt, V1, V2) → ACTIVATE(V1)
ISNAT(n__s(V1)) → ACTIVATE(V1)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
U111(tt, V1) → U121(isNatIListKind(activate(V1)), activate(V1))
U211(tt, V1) → U221(isNatKind(activate(V1)), activate(V1))
U851(tt, V2) → ISNATLIST(activate(V2))
U941(tt, L) → LENGTH(activate(L))
ISNATILIST(V) → ISNATILISTKIND(activate(V))
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
ISNATILIST(n__cons(V1, V2)) → U411(isNatKind(activate(V1)), activate(V1), activate(V2))
U931(tt, L, N) → U941(isNatKind(activate(N)), activate(L))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
U121(tt, V1) → ACTIVATE(V1)
U421(tt, V1, V2) → U431(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U851(tt, V2) → U861(isNatList(activate(V2)))
ISNATILISTKIND(n__cons(V1, V2)) → U511(isNatKind(activate(V1)), activate(V2))
U511(tt, V2) → ACTIVATE(V2)
ISNAT(n__length(V1)) → ISNATILISTKIND(activate(V1))
U311(tt, V) → ACTIVATE(V)
U841(tt, V1, V2) → ACTIVATE(V2)
U911(tt, L, N) → ISNATILISTKIND(activate(L))
U811(tt, V1, V2) → U821(isNatKind(activate(V1)), activate(V1), activate(V2))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
U811(tt, V1, V2) → ACTIVATE(V1)
U411(tt, V1, V2) → ACTIVATE(V2)
U831(tt, V1, V2) → U841(isNatIListKind(activate(V2)), activate(V1), activate(V2))
ISNATKIND(n__s(V1)) → U711(isNatKind(activate(V1)))
LENGTH(cons(N, L)) → U911(isNatList(activate(L)), activate(L), N)
U111(tt, V1) → ISNATILISTKIND(activate(V1))
U941(tt, L) → ACTIVATE(L)
LENGTH(cons(N, L)) → ACTIVATE(L)
ACTIVATE(n__length(X)) → LENGTH(activate(X))
U831(tt, V1, V2) → ACTIVATE(V2)
U321(tt, V) → U331(isNatList(activate(V)))
U441(tt, V1, V2) → ACTIVATE(V2)
U841(tt, V1, V2) → ACTIVATE(V1)
ISNATILIST(V) → U311(isNatIListKind(activate(V)), activate(V))
U851(tt, V2) → ACTIVATE(V2)
U441(tt, V1, V2) → ISNAT(activate(V1))
ZEROS → 01
ISNAT(n__length(V1)) → ACTIVATE(V1)
ACTIVATE(n__length(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof

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

U921(tt, L, N) → ACTIVATE(L)
ISNATILIST(V) → ACTIVATE(V)
U811(tt, V1, V2) → ISNATKIND(activate(V1))
ACTIVATE(n__0) → 01
U941(tt, L) → S(length(activate(L)))
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__length(V1)) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
U511(tt, V2) → ISNATILISTKIND(activate(V2))
U931(tt, L, N) → ACTIVATE(N)
ISNAT(n__length(V1)) → U111(isNatIListKind(activate(V1)), activate(V1))
ISNATKIND(n__length(V1)) → ISNATILISTKIND(activate(V1))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__s(X)) → S(activate(X))
ACTIVATE(n__cons(X1, X2)) → CONS(activate(X1), X2)
U431(tt, V1, V2) → ACTIVATE(V2)
U121(tt, V1) → U131(isNatList(activate(V1)))
U311(tt, V) → U321(isNatIListKind(activate(V)), activate(V))
U431(tt, V1, V2) → ACTIVATE(V1)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
U221(tt, V1) → U231(isNat(activate(V1)))
U451(tt, V2) → U461(isNatIList(activate(V2)))
U121(tt, V1) → ISNATLIST(activate(V1))
U831(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U911(tt, L, N) → ACTIVATE(N)
U911(tt, L, N) → ACTIVATE(L)
U921(tt, L, N) → U931(isNat(activate(N)), activate(L), activate(N))
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
U431(tt, V1, V2) → ISNATILISTKIND(activate(V2))
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V1)
U821(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U111(tt, V1) → ACTIVATE(V1)
U321(tt, V) → ISNATLIST(activate(V))
U921(tt, L, N) → ACTIVATE(N)
U911(tt, L, N) → U921(isNatIListKind(activate(L)), activate(L), activate(N))
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V2)
U821(tt, V1, V2) → U831(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U821(tt, V1, V2) → ACTIVATE(V2)
U411(tt, V1, V2) → ACTIVATE(V1)
U221(tt, V1) → ISNAT(activate(V1))
ISNATLIST(n__cons(V1, V2)) → U811(isNatKind(activate(V1)), activate(V1), activate(V2))
U221(tt, V1) → ACTIVATE(V1)
U841(tt, V1, V2) → U851(isNat(activate(V1)), activate(V2))
U841(tt, V1, V2) → ISNAT(activate(V1))
U311(tt, V) → ISNATILISTKIND(activate(V))
U511(tt, V2) → U521(isNatIListKind(activate(V2)))
U211(tt, V1) → ACTIVATE(V1)
U321(tt, V) → ACTIVATE(V)
ZEROS → CONS(0, n__zeros)
U421(tt, V1, V2) → ACTIVATE(V2)
U441(tt, V1, V2) → ACTIVATE(V1)
U431(tt, V1, V2) → U441(isNatIListKind(activate(V2)), activate(V1), activate(V2))
ISNATKIND(n__length(V1)) → U611(isNatIListKind(activate(V1)))
U211(tt, V1) → ISNATKIND(activate(V1))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
U451(tt, V2) → ACTIVATE(V2)
ACTIVATE(n__s(X)) → ACTIVATE(X)
U411(tt, V1, V2) → ISNATKIND(activate(V1))
U451(tt, V2) → ISNATILIST(activate(V2))
U821(tt, V1, V2) → ACTIVATE(V1)
U411(tt, V1, V2) → U421(isNatKind(activate(V1)), activate(V1), activate(V2))
U421(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U831(tt, V1, V2) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILISTKIND(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATLIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ACTIVATE(n__zeros) → ZEROS
LENGTH(nil) → 01
U931(tt, L, N) → ISNATKIND(activate(N))
U811(tt, V1, V2) → ACTIVATE(V2)
U921(tt, L, N) → ISNAT(activate(N))
U441(tt, V1, V2) → U451(isNat(activate(V1)), activate(V2))
U931(tt, L, N) → ACTIVATE(L)
ACTIVATE(n__nil) → NIL
U421(tt, V1, V2) → ACTIVATE(V1)
ISNAT(n__s(V1)) → ACTIVATE(V1)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
U111(tt, V1) → U121(isNatIListKind(activate(V1)), activate(V1))
U211(tt, V1) → U221(isNatKind(activate(V1)), activate(V1))
U851(tt, V2) → ISNATLIST(activate(V2))
U941(tt, L) → LENGTH(activate(L))
ISNATILIST(V) → ISNATILISTKIND(activate(V))
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
ISNATILIST(n__cons(V1, V2)) → U411(isNatKind(activate(V1)), activate(V1), activate(V2))
U931(tt, L, N) → U941(isNatKind(activate(N)), activate(L))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
U121(tt, V1) → ACTIVATE(V1)
U421(tt, V1, V2) → U431(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U851(tt, V2) → U861(isNatList(activate(V2)))
ISNATILISTKIND(n__cons(V1, V2)) → U511(isNatKind(activate(V1)), activate(V2))
U511(tt, V2) → ACTIVATE(V2)
ISNAT(n__length(V1)) → ISNATILISTKIND(activate(V1))
U311(tt, V) → ACTIVATE(V)
U841(tt, V1, V2) → ACTIVATE(V2)
U911(tt, L, N) → ISNATILISTKIND(activate(L))
U811(tt, V1, V2) → U821(isNatKind(activate(V1)), activate(V1), activate(V2))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
U811(tt, V1, V2) → ACTIVATE(V1)
U411(tt, V1, V2) → ACTIVATE(V2)
U831(tt, V1, V2) → U841(isNatIListKind(activate(V2)), activate(V1), activate(V2))
ISNATKIND(n__s(V1)) → U711(isNatKind(activate(V1)))
LENGTH(cons(N, L)) → U911(isNatList(activate(L)), activate(L), N)
U111(tt, V1) → ISNATILISTKIND(activate(V1))
U941(tt, L) → ACTIVATE(L)
LENGTH(cons(N, L)) → ACTIVATE(L)
ACTIVATE(n__length(X)) → LENGTH(activate(X))
U831(tt, V1, V2) → ACTIVATE(V2)
U321(tt, V) → U331(isNatList(activate(V)))
U441(tt, V1, V2) → ACTIVATE(V2)
U841(tt, V1, V2) → ACTIVATE(V1)
ISNATILIST(V) → U311(isNatIListKind(activate(V)), activate(V))
U851(tt, V2) → ACTIVATE(V2)
U441(tt, V1, V2) → ISNAT(activate(V1))
ZEROS → 01
ISNAT(n__length(V1)) → ACTIVATE(V1)
ACTIVATE(n__length(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs with 41 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP

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

U921(tt, L, N) → ACTIVATE(L)
U811(tt, V1, V2) → ISNATKIND(activate(V1))
U211(tt, V1) → ISNATKIND(activate(V1))
ACTIVATE(n__s(X)) → ACTIVATE(X)
U821(tt, V1, V2) → ACTIVATE(V1)
U831(tt, 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)
U511(tt, V2) → ISNATILISTKIND(activate(V2))
ISNATLIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
U931(tt, L, N) → ACTIVATE(N)
ISNAT(n__length(V1)) → U111(isNatIListKind(activate(V1)), activate(V1))
ISNATKIND(n__length(V1)) → ISNATILISTKIND(activate(V1))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
U931(tt, L, N) → ISNATKIND(activate(N))
U811(tt, V1, V2) → ACTIVATE(V2)
U921(tt, L, N) → ISNAT(activate(N))
U931(tt, L, N) → ACTIVATE(L)
ISNAT(n__s(V1)) → ACTIVATE(V1)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
U111(tt, V1) → U121(isNatIListKind(activate(V1)), activate(V1))
U211(tt, V1) → U221(isNatKind(activate(V1)), activate(V1))
U851(tt, V2) → ISNATLIST(activate(V2))
U941(tt, L) → LENGTH(activate(L))
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
U931(tt, L, N) → U941(isNatKind(activate(N)), activate(L))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
U121(tt, V1) → ACTIVATE(V1)
U911(tt, L, N) → ACTIVATE(N)
U831(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U121(tt, V1) → ISNATLIST(activate(V1))
U911(tt, L, N) → ACTIVATE(L)
ISNATILISTKIND(n__cons(V1, V2)) → U511(isNatKind(activate(V1)), activate(V2))
U511(tt, V2) → ACTIVATE(V2)
ISNAT(n__length(V1)) → ISNATILISTKIND(activate(V1))
U921(tt, L, N) → U931(isNat(activate(N)), activate(L), activate(N))
U841(tt, V1, V2) → ACTIVATE(V2)
U911(tt, L, N) → ISNATILISTKIND(activate(L))
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
U811(tt, V1, V2) → U821(isNatKind(activate(V1)), activate(V1), activate(V2))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
U811(tt, V1, V2) → ACTIVATE(V1)
U831(tt, V1, V2) → U841(isNatIListKind(activate(V2)), activate(V1), activate(V2))
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V1)
U821(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U111(tt, V1) → ACTIVATE(V1)
U921(tt, L, N) → ACTIVATE(N)
LENGTH(cons(N, L)) → U911(isNatList(activate(L)), activate(L), N)
U911(tt, L, N) → U921(isNatIListKind(activate(L)), activate(L), activate(N))
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V2)
U111(tt, V1) → ISNATILISTKIND(activate(V1))
U941(tt, L) → ACTIVATE(L)
U821(tt, V1, V2) → U831(isNatIListKind(activate(V2)), activate(V1), activate(V2))
LENGTH(cons(N, L)) → ACTIVATE(L)
U821(tt, V1, V2) → ACTIVATE(V2)
ACTIVATE(n__length(X)) → LENGTH(activate(X))
U221(tt, V1) → ISNAT(activate(V1))
ISNATLIST(n__cons(V1, V2)) → U811(isNatKind(activate(V1)), activate(V1), activate(V2))
U841(tt, V1, V2) → U851(isNat(activate(V1)), activate(V2))
U221(tt, V1) → ACTIVATE(V1)
U831(tt, V1, V2) → ACTIVATE(V2)
U841(tt, V1, V2) → ISNAT(activate(V1))
U211(tt, V1) → ACTIVATE(V1)
U841(tt, V1, V2) → ACTIVATE(V1)
U851(tt, V2) → ACTIVATE(V2)
ISNAT(n__length(V1)) → ACTIVATE(V1)
ACTIVATE(n__length(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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


The following pairs can be oriented strictly and are deleted.


ISNATKIND(n__length(V1)) → ACTIVATE(V1)
ISNATKIND(n__length(V1)) → ISNATILISTKIND(activate(V1))
U111(tt, V1) → U121(isNatIListKind(activate(V1)), activate(V1))
ISNAT(n__length(V1)) → ISNATILISTKIND(activate(V1))
U111(tt, V1) → ACTIVATE(V1)
U111(tt, V1) → ISNATILISTKIND(activate(V1))
ACTIVATE(n__length(X)) → LENGTH(activate(X))
ISNAT(n__length(V1)) → ACTIVATE(V1)
ACTIVATE(n__length(X)) → ACTIVATE(X)
The remaining pairs can at least be oriented weakly.

U921(tt, L, N) → ACTIVATE(L)
U811(tt, V1, V2) → ISNATKIND(activate(V1))
U211(tt, V1) → ISNATKIND(activate(V1))
ACTIVATE(n__s(X)) → ACTIVATE(X)
U821(tt, V1, V2) → ACTIVATE(V1)
U831(tt, V1, V2) → ACTIVATE(V1)
ISNATILISTKIND(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
U511(tt, V2) → ISNATILISTKIND(activate(V2))
ISNATLIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
U931(tt, L, N) → ACTIVATE(N)
ISNAT(n__length(V1)) → U111(isNatIListKind(activate(V1)), activate(V1))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
U931(tt, L, N) → ISNATKIND(activate(N))
U811(tt, V1, V2) → ACTIVATE(V2)
U921(tt, L, N) → ISNAT(activate(N))
U931(tt, L, N) → ACTIVATE(L)
ISNAT(n__s(V1)) → ACTIVATE(V1)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
U211(tt, V1) → U221(isNatKind(activate(V1)), activate(V1))
U851(tt, V2) → ISNATLIST(activate(V2))
U941(tt, L) → LENGTH(activate(L))
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
U931(tt, L, N) → U941(isNatKind(activate(N)), activate(L))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
U121(tt, V1) → ACTIVATE(V1)
U911(tt, L, N) → ACTIVATE(N)
U831(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U121(tt, V1) → ISNATLIST(activate(V1))
U911(tt, L, N) → ACTIVATE(L)
ISNATILISTKIND(n__cons(V1, V2)) → U511(isNatKind(activate(V1)), activate(V2))
U511(tt, V2) → ACTIVATE(V2)
U921(tt, L, N) → U931(isNat(activate(N)), activate(L), activate(N))
U841(tt, V1, V2) → ACTIVATE(V2)
U911(tt, L, N) → ISNATILISTKIND(activate(L))
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
U811(tt, V1, V2) → U821(isNatKind(activate(V1)), activate(V1), activate(V2))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
U811(tt, V1, V2) → ACTIVATE(V1)
U831(tt, V1, V2) → U841(isNatIListKind(activate(V2)), activate(V1), activate(V2))
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V1)
U821(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U921(tt, L, N) → ACTIVATE(N)
LENGTH(cons(N, L)) → U911(isNatList(activate(L)), activate(L), N)
U911(tt, L, N) → U921(isNatIListKind(activate(L)), activate(L), activate(N))
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V2)
U941(tt, L) → ACTIVATE(L)
U821(tt, V1, V2) → U831(isNatIListKind(activate(V2)), activate(V1), activate(V2))
LENGTH(cons(N, L)) → ACTIVATE(L)
U821(tt, V1, V2) → ACTIVATE(V2)
U221(tt, V1) → ISNAT(activate(V1))
ISNATLIST(n__cons(V1, V2)) → U811(isNatKind(activate(V1)), activate(V1), activate(V2))
U841(tt, V1, V2) → U851(isNat(activate(V1)), activate(V2))
U221(tt, V1) → ACTIVATE(V1)
U831(tt, V1, V2) → ACTIVATE(V2)
U841(tt, V1, V2) → ISNAT(activate(V1))
U211(tt, V1) → ACTIVATE(V1)
U841(tt, V1, V2) → ACTIVATE(V1)
U851(tt, V2) → ACTIVATE(V2)
Used ordering: Polynomial interpretation [25]:

POL(0) = 0   
POL(ACTIVATE(x1)) = x1   
POL(ISNAT(x1)) = x1   
POL(ISNATILISTKIND(x1)) = x1   
POL(ISNATKIND(x1)) = x1   
POL(ISNATLIST(x1)) = x1   
POL(LENGTH(x1)) = x1   
POL(U11(x1, x2)) = 0   
POL(U111(x1, x2)) = 1 + x2   
POL(U12(x1, x2)) = 0   
POL(U121(x1, x2)) = x2   
POL(U13(x1)) = 0   
POL(U21(x1, x2)) = 0   
POL(U211(x1, x2)) = x2   
POL(U22(x1, x2)) = 0   
POL(U221(x1, x2)) = x2   
POL(U23(x1)) = 0   
POL(U51(x1, x2)) = 0   
POL(U511(x1, x2)) = x2   
POL(U52(x1)) = 0   
POL(U61(x1)) = 0   
POL(U71(x1)) = 0   
POL(U81(x1, x2, x3)) = x3   
POL(U811(x1, x2, x3)) = x2 + x3   
POL(U82(x1, x2, x3)) = 0   
POL(U821(x1, x2, x3)) = x2 + x3   
POL(U83(x1, x2, x3)) = 0   
POL(U831(x1, x2, x3)) = x2 + x3   
POL(U84(x1, x2, x3)) = 0   
POL(U841(x1, x2, x3)) = x2 + x3   
POL(U85(x1, x2)) = 0   
POL(U851(x1, x2)) = x2   
POL(U86(x1)) = 0   
POL(U91(x1, x2, x3)) = 1 + x2   
POL(U911(x1, x2, x3)) = x2 + x3   
POL(U92(x1, x2, x3)) = 1 + x2   
POL(U921(x1, x2, x3)) = x2 + x3   
POL(U93(x1, x2, x3)) = 1 + x2   
POL(U931(x1, x2, x3)) = x2 + x3   
POL(U94(x1, x2)) = 1 + x2   
POL(U941(x1, x2)) = x2   
POL(activate(x1)) = x1   
POL(cons(x1, x2)) = x1 + x2   
POL(isNat(x1)) = 0   
POL(isNatIListKind(x1)) = 0   
POL(isNatKind(x1)) = 0   
POL(isNatList(x1)) = x1   
POL(length(x1)) = 1 + x1   
POL(n__0) = 0   
POL(n__cons(x1, x2)) = x1 + x2   
POL(n__length(x1)) = 1 + x1   
POL(n__nil) = 1   
POL(n__s(x1)) = x1   
POL(n__zeros) = 0   
POL(nil) = 1   
POL(s(x1)) = x1   
POL(tt) = 0   
POL(zeros) = 0   

The following usable rules [17] were oriented:

nil → n__nil
activate(n__zeros) → zeros
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__nil) → nil
activate(X) → X
zeros → cons(0, n__zeros)
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
U94(tt, L) → s(length(activate(L)))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U86(tt) → tt
U85(tt, V2) → U86(isNatList(activate(V2)))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U71(tt) → tt
U61(tt) → tt
U52(tt) → tt
0 → n__0
length(X) → n__length(X)
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ DependencyGraphProof
          ↳ QDP

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

U921(tt, L, N) → ACTIVATE(L)
U811(tt, V1, V2) → ISNATKIND(activate(V1))
U211(tt, V1) → ISNATKIND(activate(V1))
ACTIVATE(n__s(X)) → ACTIVATE(X)
U821(tt, V1, V2) → ACTIVATE(V1)
U831(tt, V1, V2) → ACTIVATE(V1)
ISNATILISTKIND(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
U511(tt, V2) → ISNATILISTKIND(activate(V2))
ISNATLIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
U931(tt, L, N) → ACTIVATE(N)
ISNAT(n__length(V1)) → U111(isNatIListKind(activate(V1)), activate(V1))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
U931(tt, L, N) → ISNATKIND(activate(N))
U811(tt, V1, V2) → ACTIVATE(V2)
U921(tt, L, N) → ISNAT(activate(N))
U931(tt, L, N) → ACTIVATE(L)
ISNAT(n__s(V1)) → ACTIVATE(V1)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
U211(tt, V1) → U221(isNatKind(activate(V1)), activate(V1))
U851(tt, V2) → ISNATLIST(activate(V2))
U941(tt, L) → LENGTH(activate(L))
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
U931(tt, L, N) → U941(isNatKind(activate(N)), activate(L))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
U121(tt, V1) → ACTIVATE(V1)
U911(tt, L, N) → ACTIVATE(N)
U831(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U121(tt, V1) → ISNATLIST(activate(V1))
U911(tt, L, N) → ACTIVATE(L)
ISNATILISTKIND(n__cons(V1, V2)) → U511(isNatKind(activate(V1)), activate(V2))
U511(tt, V2) → ACTIVATE(V2)
U921(tt, L, N) → U931(isNat(activate(N)), activate(L), activate(N))
U841(tt, V1, V2) → ACTIVATE(V2)
U911(tt, L, N) → ISNATILISTKIND(activate(L))
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
U811(tt, V1, V2) → U821(isNatKind(activate(V1)), activate(V1), activate(V2))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
U811(tt, V1, V2) → ACTIVATE(V1)
U831(tt, V1, V2) → U841(isNatIListKind(activate(V2)), activate(V1), activate(V2))
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V1)
U821(tt, V1, V2) → ISNATILISTKIND(activate(V2))
U921(tt, L, N) → ACTIVATE(N)
LENGTH(cons(N, L)) → U911(isNatList(activate(L)), activate(L), N)
U911(tt, L, N) → U921(isNatIListKind(activate(L)), activate(L), activate(N))
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V2)
U941(tt, L) → ACTIVATE(L)
U821(tt, V1, V2) → U831(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U821(tt, V1, V2) → ACTIVATE(V2)
LENGTH(cons(N, L)) → ACTIVATE(L)
U221(tt, V1) → ISNAT(activate(V1))
ISNATLIST(n__cons(V1, V2)) → U811(isNatKind(activate(V1)), activate(V1), activate(V2))
U221(tt, V1) → ACTIVATE(V1)
U841(tt, V1, V2) → U851(isNat(activate(V1)), activate(V2))
U841(tt, V1, V2) → ISNAT(activate(V1))
U831(tt, V1, V2) → ACTIVATE(V2)
U211(tt, V1) → ACTIVATE(V1)
U841(tt, V1, V2) → ACTIVATE(V1)
U851(tt, V2) → ACTIVATE(V2)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 6 SCCs with 41 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ DependencyGraphProof
                  ↳ AND
                    ↳ QDP
                      ↳ UsableRulesProof
                    ↳ QDP
                    ↳ QDP
                    ↳ QDP
                    ↳ QDP
                    ↳ QDP
          ↳ QDP

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

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

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ DependencyGraphProof
                  ↳ AND
                    ↳ QDP
                      ↳ UsableRulesProof
                        ↳ QDP
                          ↳ QDPSizeChangeProof
                    ↳ QDP
                    ↳ QDP
                    ↳ QDP
                    ↳ QDP
                    ↳ QDP
          ↳ QDP

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

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

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

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



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ DependencyGraphProof
                  ↳ AND
                    ↳ QDP
                    ↳ QDP
                      ↳ QDPOrderProof
                    ↳ QDP
                    ↳ QDP
                    ↳ QDP
                    ↳ QDP
          ↳ QDP

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

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

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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


The following pairs can be oriented strictly and are deleted.


ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
The remaining pairs can at least be oriented weakly.
none
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
M( U82(x1, ..., x3) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/00\
\10/
·x3

M( U12(x1, x2) ) =
/0\
\1/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( activate(x1) ) =
/0\
\0/
+
/10\
\01/
·x1

M( n__nil ) =
/1\
\0/

M( n__s(x1) ) =
/1\
\1/
+
/10\
\00/
·x1

M( U21(x1, x2) ) =
/0\
\1/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( isNatIListKind(x1) ) =
/1\
\1/
+
/00\
\00/
·x1

M( U51(x1, x2) ) =
/0\
\1/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( U23(x1) ) =
/0\
\1/
+
/00\
\00/
·x1

M( tt ) =
/0\
\1/

M( U93(x1, ..., x3) ) =
/1\
\1/
+
/00\
\00/
·x1+
/01\
\00/
·x2+
/00\
\00/
·x3

M( isNatList(x1) ) =
/0\
\0/
+
/00\
\10/
·x1

M( U52(x1) ) =
/0\
\1/
+
/00\
\00/
·x1

M( zeros ) =
/0\
\0/

M( U85(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\10/
·x2

M( s(x1) ) =
/1\
\1/
+
/10\
\00/
·x1

M( isNat(x1) ) =
/0\
\0/
+
/00\
\01/
·x1

M( U71(x1) ) =
/0\
\1/
+
/00\
\00/
·x1

M( nil ) =
/1\
\0/

M( U61(x1) ) =
/0\
\1/
+
/00\
\10/
·x1

M( U83(x1, ..., x3) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/00\
\10/
·x3

M( n__length(x1) ) =
/0\
\1/
+
/01\
\00/
·x1

M( U22(x1, x2) ) =
/0\
\1/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( U81(x1, ..., x3) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/00\
\10/
·x3

M( n__zeros ) =
/0\
\0/

M( n__cons(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/10\
\11/
·x2

M( U11(x1, x2) ) =
/0\
\1/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( isNatKind(x1) ) =
/0\
\1/
+
/00\
\01/
·x1

M( 0 ) =
/0\
\0/

M( U94(x1, x2) ) =
/1\
\1/
+
/00\
\00/
·x1+
/01\
\00/
·x2

M( cons(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/10\
\11/
·x2

M( U91(x1, ..., x3) ) =
/0\
\1/
+
/01\
\00/
·x1+
/01\
\00/
·x2+
/00\
\00/
·x3

M( n__0 ) =
/0\
\0/

M( U13(x1) ) =
/0\
\1/
+
/00\
\00/
·x1

M( U86(x1) ) =
/0\
\0/
+
/00\
\01/
·x1

M( U92(x1, ..., x3) ) =
/1\
\1/
+
/00\
\00/
·x1+
/01\
\00/
·x2+
/00\
\00/
·x3

M( U84(x1, ..., x3) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/00\
\10/
·x3

M( length(x1) ) =
/0\
\1/
+
/01\
\00/
·x1

Tuple symbols:
M( ISNATKIND(x1) ) = 0+
[1,0]
·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)
nil → n__nil
activate(n__zeros) → zeros
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
activate(n__nil) → nil
activate(X) → X
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
U94(tt, L) → s(length(activate(L)))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U86(tt) → tt
U85(tt, V2) → U86(isNatList(activate(V2)))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U71(tt) → tt
U61(tt) → tt
U52(tt) → tt
0 → n__0
length(X) → n__length(X)
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__nil) → tt



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ DependencyGraphProof
                  ↳ AND
                    ↳ QDP
                    ↳ QDP
                      ↳ QDPOrderProof
                        ↳ QDP
                          ↳ PisEmptyProof
                    ↳ QDP
                    ↳ QDP
                    ↳ QDP
                    ↳ QDP
          ↳ QDP

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

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ DependencyGraphProof
                  ↳ AND
                    ↳ QDP
                    ↳ QDP
                    ↳ QDP
                      ↳ Narrowing
                    ↳ QDP
                    ↳ QDP
                    ↳ QDP
          ↳ QDP

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

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

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

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



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ DependencyGraphProof
                  ↳ AND
                    ↳ QDP
                    ↳ QDP
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ Narrowing
                    ↳ QDP
                    ↳ QDP
                    ↳ QDP
          ↳ QDP

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

U511(tt, n__s(x0)) → ISNATILISTKIND(s(activate(x0)))
U511(tt, n__0) → ISNATILISTKIND(0)
U511(tt, n__length(x0)) → ISNATILISTKIND(length(activate(x0)))
U511(tt, x0) → ISNATILISTKIND(x0)
U511(tt, n__nil) → ISNATILISTKIND(nil)
ISNATILISTKIND(n__cons(V1, V2)) → U511(isNatKind(activate(V1)), activate(V2))
U511(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
U511(tt, n__zeros) → ISNATILISTKIND(zeros)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

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



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ DependencyGraphProof
                  ↳ AND
                    ↳ QDP
                    ↳ QDP
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ Narrowing
                    ↳ QDP
                    ↳ QDP
                    ↳ QDP
          ↳ QDP

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

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

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

U511(tt, n__0) → ISNATILISTKIND(n__0)



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ DependencyGraphProof
                  ↳ AND
                    ↳ QDP
                    ↳ QDP
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                    ↳ QDP
                    ↳ QDP
                    ↳ QDP
          ↳ QDP

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

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

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ DependencyGraphProof
                  ↳ AND
                    ↳ QDP
                    ↳ QDP
                    ↳ 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:

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

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

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



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ DependencyGraphProof
                  ↳ AND
                    ↳ QDP
                    ↳ QDP
                    ↳ 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:

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

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

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



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ DependencyGraphProof
                  ↳ AND
                    ↳ QDP
                    ↳ QDP
                    ↳ 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:

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

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ DependencyGraphProof
                  ↳ AND
                    ↳ QDP
                    ↳ QDP
                    ↳ 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__s(x0)) → ISNATILISTKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U511(isNatKind(s(activate(x0))), activate(y1))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U511(isNatKind(cons(activate(x0), x1)), activate(y1))
U511(tt, n__length(x0)) → ISNATILISTKIND(length(activate(x0)))
U511(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(n__zeros, y1)) → U511(isNatKind(zeros), activate(y1))
U511(tt, n__nil) → ISNATILISTKIND(nil)
ISNATILISTKIND(n__cons(n__length(x0), y1)) → U511(isNatKind(length(activate(x0))), activate(y1))
ISNATILISTKIND(n__cons(x0, y1)) → U511(isNatKind(x0), activate(y1))
ISNATILISTKIND(n__cons(n__0, y0)) → U511(isNatKind(n__0), activate(y0))
U511(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
U511(tt, n__zeros) → ISNATILISTKIND(zeros)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

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



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ DependencyGraphProof
                  ↳ AND
                    ↳ QDP
                    ↳ QDP
                    ↳ 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:

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

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ DependencyGraphProof
                  ↳ AND
                    ↳ QDP
                    ↳ QDP
                    ↳ 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__s(x0)) → ISNATILISTKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U511(isNatKind(s(activate(x0))), activate(y1))
ISNATILISTKIND(n__cons(n__zeros, y0)) → U511(isNatKind(cons(0, n__zeros)), activate(y0))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U511(isNatKind(cons(activate(x0), x1)), activate(y1))
U511(tt, n__length(x0)) → ISNATILISTKIND(length(activate(x0)))
U511(tt, x0) → ISNATILISTKIND(x0)
U511(tt, n__nil) → ISNATILISTKIND(nil)
ISNATILISTKIND(n__cons(n__length(x0), y1)) → U511(isNatKind(length(activate(x0))), activate(y1))
ISNATILISTKIND(n__cons(x0, y1)) → U511(isNatKind(x0), activate(y1))
ISNATILISTKIND(n__cons(n__0, y0)) → U511(isNatKind(n__0), activate(y0))
U511(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
U511(tt, n__zeros) → ISNATILISTKIND(zeros)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

U511(tt, n__nil) → ISNATILISTKIND(n__nil)



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ DependencyGraphProof
                  ↳ AND
                    ↳ QDP
                    ↳ QDP
                    ↳ 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:

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

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ DependencyGraphProof
                  ↳ AND
                    ↳ QDP
                    ↳ QDP
                    ↳ 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__s(x0)) → ISNATILISTKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U511(isNatKind(s(activate(x0))), activate(y1))
ISNATILISTKIND(n__cons(n__zeros, y0)) → U511(isNatKind(cons(0, n__zeros)), activate(y0))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U511(isNatKind(cons(activate(x0), x1)), activate(y1))
U511(tt, n__length(x0)) → ISNATILISTKIND(length(activate(x0)))
U511(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(n__length(x0), y1)) → U511(isNatKind(length(activate(x0))), activate(y1))
ISNATILISTKIND(n__cons(x0, y1)) → U511(isNatKind(x0), activate(y1))
ISNATILISTKIND(n__cons(n__0, y0)) → U511(isNatKind(n__0), activate(y0))
U511(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
U511(tt, n__zeros) → ISNATILISTKIND(zeros)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

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



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ DependencyGraphProof
                  ↳ AND
                    ↳ QDP
                    ↳ QDP
                    ↳ 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:

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

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ DependencyGraphProof
                  ↳ AND
                    ↳ QDP
                    ↳ QDP
                    ↳ 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__s(x0)) → ISNATILISTKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U511(isNatKind(s(activate(x0))), activate(y1))
ISNATILISTKIND(n__cons(n__zeros, y0)) → U511(isNatKind(cons(0, n__zeros)), activate(y0))
U511(tt, n__zeros) → ISNATILISTKIND(cons(0, n__zeros))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U511(isNatKind(cons(activate(x0), x1)), activate(y1))
U511(tt, n__length(x0)) → ISNATILISTKIND(length(activate(x0)))
U511(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(n__length(x0), y1)) → U511(isNatKind(length(activate(x0))), activate(y1))
ISNATILISTKIND(n__cons(x0, y1)) → U511(isNatKind(x0), activate(y1))
ISNATILISTKIND(n__cons(n__0, y0)) → U511(isNatKind(n__0), activate(y0))
U511(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

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



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ DependencyGraphProof
                  ↳ AND
                    ↳ QDP
                    ↳ QDP
                    ↳ 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
                                                                              ↳ Narrowing
                    ↳ QDP
                    ↳ QDP
                    ↳ QDP
          ↳ QDP

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

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

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

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



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ DependencyGraphProof
                  ↳ AND
                    ↳ QDP
                    ↳ QDP
                    ↳ 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
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                    ↳ QDP
                    ↳ QDP
                    ↳ QDP
          ↳ QDP

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

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

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ DependencyGraphProof
                  ↳ AND
                    ↳ QDP
                    ↳ QDP
                    ↳ 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
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                    ↳ QDP
                    ↳ QDP
                    ↳ QDP
          ↳ QDP

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

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

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

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



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ DependencyGraphProof
                  ↳ AND
                    ↳ QDP
                    ↳ QDP
                    ↳ 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
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                    ↳ QDP
                    ↳ QDP
                    ↳ QDP
          ↳ QDP

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

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

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ DependencyGraphProof
                  ↳ AND
                    ↳ QDP
                    ↳ QDP
                    ↳ 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
                                                                              ↳ 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__zeros) → ISNATILISTKIND(n__cons(0, n__zeros))
U511(tt, n__s(x0)) → ISNATILISTKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U511(isNatKind(s(activate(x0))), activate(y1))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U511(isNatKind(cons(activate(x0), x1)), activate(y1))
U511(tt, n__length(x0)) → ISNATILISTKIND(length(activate(x0)))
U511(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(n__length(x0), y1)) → U511(isNatKind(length(activate(x0))), activate(y1))
ISNATILISTKIND(n__cons(x0, y1)) → U511(isNatKind(x0), activate(y1))
U511(tt, n__zeros) → ISNATILISTKIND(cons(n__0, n__zeros))
ISNATILISTKIND(n__cons(n__0, y0)) → U511(isNatKind(n__0), activate(y0))
U511(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

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



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ DependencyGraphProof
                  ↳ AND
                    ↳ QDP
                    ↳ QDP
                    ↳ 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
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ QDPOrderProof
                    ↳ QDP
                    ↳ QDP
                    ↳ QDP
          ↳ QDP

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

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

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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


The following pairs can be oriented strictly and are deleted.


ISNATILISTKIND(n__cons(n__length(x0), y1)) → U511(isNatKind(length(activate(x0))), activate(y1))
The remaining pairs can at least be oriented weakly.

U511(tt, n__s(x0)) → ISNATILISTKIND(s(activate(x0)))
U511(tt, n__zeros) → ISNATILISTKIND(n__cons(0, n__zeros))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U511(isNatKind(s(activate(x0))), activate(y1))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U511(isNatKind(cons(activate(x0), x1)), activate(y1))
U511(tt, n__length(x0)) → ISNATILISTKIND(length(activate(x0)))
U511(tt, x0) → ISNATILISTKIND(x0)
U511(tt, n__zeros) → ISNATILISTKIND(n__cons(n__0, n__zeros))
ISNATILISTKIND(n__cons(x0, y1)) → U511(isNatKind(x0), activate(y1))
ISNATILISTKIND(n__cons(n__0, y0)) → U511(isNatKind(n__0), activate(y0))
U511(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
Used ordering: Polynomial interpretation [25]:

POL(0) = 0   
POL(ISNATILISTKIND(x1)) = x1   
POL(U11(x1, x2)) = 1   
POL(U12(x1, x2)) = 1   
POL(U13(x1)) = 1   
POL(U21(x1, x2)) = 1   
POL(U22(x1, x2)) = 0   
POL(U23(x1)) = 0   
POL(U51(x1, x2)) = x2   
POL(U511(x1, x2)) = x2   
POL(U52(x1)) = 0   
POL(U61(x1)) = 0   
POL(U71(x1)) = 0   
POL(U81(x1, x2, x3)) = 0   
POL(U82(x1, x2, x3)) = 0   
POL(U83(x1, x2, x3)) = 0   
POL(U84(x1, x2, x3)) = 0   
POL(U85(x1, x2)) = 0   
POL(U86(x1)) = 0   
POL(U91(x1, x2, x3)) = 0   
POL(U92(x1, x2, x3)) = 0   
POL(U93(x1, x2, x3)) = 0   
POL(U94(x1, x2)) = 0   
POL(activate(x1)) = x1   
POL(cons(x1, x2)) = x1 + x2   
POL(isNat(x1)) = 1 + x1   
POL(isNatIListKind(x1)) = x1   
POL(isNatKind(x1)) = 0   
POL(isNatList(x1)) = 0   
POL(length(x1)) = 1   
POL(n__0) = 0   
POL(n__cons(x1, x2)) = x1 + x2   
POL(n__length(x1)) = 1   
POL(n__nil) = 0   
POL(n__s(x1)) = 0   
POL(n__zeros) = 0   
POL(nil) = 0   
POL(s(x1)) = 0   
POL(tt) = 0   
POL(zeros) = 0   

The following usable rules [17] were oriented:

zeros → cons(0, n__zeros)
nil → n__nil
activate(n__zeros) → zeros
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
activate(n__nil) → nil
activate(X) → X
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
U94(tt, L) → s(length(activate(L)))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U86(tt) → tt
U85(tt, V2) → U86(isNatList(activate(V2)))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U71(tt) → tt
U61(tt) → tt
U52(tt) → tt
0 → n__0
length(X) → n__length(X)
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ DependencyGraphProof
                  ↳ AND
                    ↳ QDP
                    ↳ QDP
                    ↳ 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
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ QDPOrderProof
                                                                                                    ↳ QDP
                                                                                                      ↳ QDPOrderProof
                    ↳ QDP
                    ↳ QDP
                    ↳ QDP
          ↳ QDP

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

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

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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


The following pairs can be oriented strictly and are deleted.


ISNATILISTKIND(n__cons(n__s(x0), y1)) → U511(isNatKind(s(activate(x0))), activate(y1))
The remaining pairs can at least be oriented weakly.

U511(tt, n__zeros) → ISNATILISTKIND(n__cons(0, n__zeros))
U511(tt, n__s(x0)) → ISNATILISTKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U511(isNatKind(cons(activate(x0), x1)), activate(y1))
U511(tt, n__length(x0)) → ISNATILISTKIND(length(activate(x0)))
U511(tt, x0) → ISNATILISTKIND(x0)
U511(tt, n__zeros) → ISNATILISTKIND(n__cons(n__0, n__zeros))
ISNATILISTKIND(n__cons(x0, y1)) → U511(isNatKind(x0), activate(y1))
ISNATILISTKIND(n__cons(n__0, y0)) → U511(isNatKind(n__0), activate(y0))
U511(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
Used ordering: Polynomial interpretation [25]:

POL(0) = 0   
POL(ISNATILISTKIND(x1)) = x1   
POL(U11(x1, x2)) = x2   
POL(U12(x1, x2)) = x2   
POL(U13(x1)) = x1   
POL(U21(x1, x2)) = 1   
POL(U22(x1, x2)) = 1   
POL(U23(x1)) = 1   
POL(U51(x1, x2)) = 1   
POL(U511(x1, x2)) = x2   
POL(U52(x1)) = 1   
POL(U61(x1)) = x1   
POL(U71(x1)) = 1   
POL(U81(x1, x2, x3)) = x2 + x3   
POL(U82(x1, x2, x3)) = x2 + x3   
POL(U83(x1, x2, x3)) = x2 + x3   
POL(U84(x1, x2, x3)) = x3   
POL(U85(x1, x2)) = x2   
POL(U86(x1)) = x1   
POL(U91(x1, x2, x3)) = x1 + x3   
POL(U92(x1, x2, x3)) = x1 + x3   
POL(U93(x1, x2, x3)) = x1   
POL(U94(x1, x2)) = 1   
POL(activate(x1)) = x1   
POL(cons(x1, x2)) = x1 + x2   
POL(isNat(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__cons(x1, x2)) = x1 + x2   
POL(n__length(x1)) = x1   
POL(n__nil) = 1   
POL(n__s(x1)) = 1   
POL(n__zeros) = 0   
POL(nil) = 1   
POL(s(x1)) = 1   
POL(tt) = 1   
POL(zeros) = 0   

The following usable rules [17] were oriented:

zeros → cons(0, n__zeros)
nil → n__nil
activate(n__zeros) → zeros
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
activate(n__nil) → nil
activate(X) → X
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__0) → tt
U94(tt, L) → s(length(activate(L)))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U86(tt) → tt
U85(tt, V2) → U86(isNatList(activate(V2)))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U71(tt) → tt
U61(tt) → tt
U52(tt) → tt
0 → n__0
length(X) → n__length(X)
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatIListKind(n__nil) → tt



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ DependencyGraphProof
                  ↳ AND
                    ↳ QDP
                    ↳ QDP
                    ↳ 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
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ QDPOrderProof
                                                                                                    ↳ QDP
                                                                                                      ↳ QDPOrderProof
                                                                                                        ↳ QDP
                                                                                                          ↳ QDPOrderProof
                    ↳ QDP
                    ↳ QDP
                    ↳ QDP
          ↳ QDP

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

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

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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


The following pairs can be oriented strictly and are deleted.


ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U511(isNatKind(cons(activate(x0), x1)), activate(y1))
The remaining pairs can at least be oriented weakly.

U511(tt, n__s(x0)) → ISNATILISTKIND(s(activate(x0)))
U511(tt, n__zeros) → ISNATILISTKIND(n__cons(0, n__zeros))
U511(tt, n__length(x0)) → ISNATILISTKIND(length(activate(x0)))
U511(tt, x0) → ISNATILISTKIND(x0)
U511(tt, n__zeros) → ISNATILISTKIND(n__cons(n__0, n__zeros))
ISNATILISTKIND(n__cons(x0, y1)) → U511(isNatKind(x0), activate(y1))
ISNATILISTKIND(n__cons(n__0, y0)) → U511(isNatKind(n__0), activate(y0))
U511(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
M( U82(x1, ..., x3) ) =
/1\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/10\
\00/
·x3

M( U12(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( activate(x1) ) =
/0\
\0/
+
/10\
\01/
·x1

M( n__nil ) =
/0\
\0/

M( n__s(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( U21(x1, x2) ) =
/1\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( isNatIListKind(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( U51(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( U23(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( tt ) =
/0\
\0/

M( U93(x1, ..., x3) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/00\
\01/
·x3

M( isNatList(x1) ) =
/1\
\0/
+
/10\
\00/
·x1

M( U52(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( zeros ) =
/0\
\1/

M( U85(x1, x2) ) =
/1\
\0/
+
/00\
\00/
·x1+
/10\
\00/
·x2

M( s(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( isNat(x1) ) =
/1\
\0/
+
/00\
\00/
·x1

M( U71(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( nil ) =
/0\
\0/

M( U61(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( U83(x1, ..., x3) ) =
/1\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/10\
\00/
·x3

M( n__length(x1) ) =
/0\
\0/
+
/00\
\10/
·x1

M( U22(x1, x2) ) =
/1\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( U81(x1, ..., x3) ) =
/1\
\0/
+
/00\
\00/
·x1+
/01\
\00/
·x2+
/10\
\00/
·x3

M( n__zeros ) =
/0\
\1/

M( n__cons(x1, x2) ) =
/0\
\1/
+
/01\
\00/
·x1+
/10\
\00/
·x2

M( U11(x1, x2) ) =
/1\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( isNatKind(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( 0 ) =
/0\
\0/

M( U94(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( cons(x1, x2) ) =
/0\
\1/
+
/01\
\00/
·x1+
/10\
\00/
·x2

M( U91(x1, ..., x3) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/00\
\01/
·x3

M( n__0 ) =
/0\
\0/

M( U13(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( U86(x1) ) =
/1\
\0/
+
/00\
\00/
·x1

M( U92(x1, ..., x3) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/00\
\01/
·x3

M( U84(x1, ..., x3) ) =
/1\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/10\
\00/
·x3

M( length(x1) ) =
/0\
\0/
+
/00\
\10/
·x1

Tuple symbols:
M( U511(x1, x2) ) = 0+
[0,0]
·x1+
[1,0]
·x2

M( ISNATILISTKIND(x1) ) = 0+
[1,0]
·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)
nil → n__nil
activate(n__zeros) → zeros
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
activate(n__nil) → nil
activate(X) → X
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__0) → tt
U94(tt, L) → s(length(activate(L)))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U86(tt) → tt
U85(tt, V2) → U86(isNatList(activate(V2)))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U71(tt) → tt
U61(tt) → tt
U52(tt) → tt
0 → n__0
length(X) → n__length(X)
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ DependencyGraphProof
                  ↳ AND
                    ↳ QDP
                    ↳ QDP
                    ↳ 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
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ 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, n__zeros) → ISNATILISTKIND(n__cons(0, n__zeros))
U511(tt, n__s(x0)) → ISNATILISTKIND(s(activate(x0)))
U511(tt, n__length(x0)) → ISNATILISTKIND(length(activate(x0)))
U511(tt, x0) → ISNATILISTKIND(x0)
U511(tt, n__zeros) → ISNATILISTKIND(n__cons(n__0, n__zeros))
ISNATILISTKIND(n__cons(x0, y1)) → U511(isNatKind(x0), activate(y1))
ISNATILISTKIND(n__cons(n__0, y0)) → U511(isNatKind(n__0), activate(y0))
U511(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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


The following pairs can be oriented strictly and are deleted.


U511(tt, n__length(x0)) → ISNATILISTKIND(length(activate(x0)))
The remaining pairs can at least be oriented weakly.

U511(tt, n__zeros) → ISNATILISTKIND(n__cons(0, n__zeros))
U511(tt, n__s(x0)) → ISNATILISTKIND(s(activate(x0)))
U511(tt, x0) → ISNATILISTKIND(x0)
U511(tt, n__zeros) → ISNATILISTKIND(n__cons(n__0, n__zeros))
ISNATILISTKIND(n__cons(x0, y1)) → U511(isNatKind(x0), activate(y1))
ISNATILISTKIND(n__cons(n__0, y0)) → U511(isNatKind(n__0), activate(y0))
U511(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
M( U82(x1, ..., x3) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/00\
\00/
·x3

M( U12(x1, x2) ) =
/0\
\1/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( activate(x1) ) =
/0\
\0/
+
/10\
\01/
·x1

M( n__nil ) =
/0\
\0/

M( n__s(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( U21(x1, x2) ) =
/0\
\1/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( isNatIListKind(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( U51(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( U23(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( tt ) =
/0\
\0/

M( U93(x1, ..., x3) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/00\
\00/
·x3

M( isNatList(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( U52(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( zeros ) =
/0\
\0/

M( U85(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( s(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( isNat(x1) ) =
/0\
\1/
+
/00\
\00/
·x1

M( U71(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( nil ) =
/0\
\0/

M( U61(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( U83(x1, ..., x3) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/00\
\00/
·x3

M( n__length(x1) ) =
/1\
\0/
+
/00\
\00/
·x1

M( U22(x1, x2) ) =
/0\
\1/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( U81(x1, ..., x3) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/00\
\00/
·x3

M( n__zeros ) =
/0\
\0/

M( n__cons(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\11/
·x2

M( U11(x1, x2) ) =
/0\
\1/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( isNatKind(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( 0 ) =
/0\
\0/

M( U94(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( cons(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\11/
·x2

M( U91(x1, ..., x3) ) =
/1\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/00\
\00/
·x3

M( n__0 ) =
/0\
\0/

M( U13(x1) ) =
/0\
\1/
+
/00\
\00/
·x1

M( U86(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( U92(x1, ..., x3) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/00\
\00/
·x3

M( U84(x1, ..., x3) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/00\
\00/
·x3

M( length(x1) ) =
/1\
\0/
+
/00\
\00/
·x1

Tuple symbols:
M( U511(x1, x2) ) = 0+
[0,0]
·x1+
[1,1]
·x2

M( ISNATILISTKIND(x1) ) = 0+
[0,1]
·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)
nil → n__nil
activate(n__zeros) → zeros
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
activate(n__nil) → nil
activate(X) → X
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__0) → tt
U94(tt, L) → s(length(activate(L)))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U86(tt) → tt
U85(tt, V2) → U86(isNatList(activate(V2)))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U71(tt) → tt
U61(tt) → tt
U52(tt) → tt
0 → n__0
length(X) → n__length(X)
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ DependencyGraphProof
                  ↳ AND
                    ↳ QDP
                    ↳ QDP
                    ↳ 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
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ 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, n__s(x0)) → ISNATILISTKIND(s(activate(x0)))
U511(tt, n__zeros) → ISNATILISTKIND(n__cons(0, n__zeros))
U511(tt, x0) → ISNATILISTKIND(x0)
U511(tt, n__zeros) → ISNATILISTKIND(n__cons(n__0, n__zeros))
ISNATILISTKIND(n__cons(x0, y1)) → U511(isNatKind(x0), activate(y1))
ISNATILISTKIND(n__cons(n__0, y0)) → U511(isNatKind(n__0), activate(y0))
U511(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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


The following pairs can be oriented strictly and are deleted.


U511(tt, n__s(x0)) → ISNATILISTKIND(s(activate(x0)))
The remaining pairs can at least be oriented weakly.

U511(tt, n__zeros) → ISNATILISTKIND(n__cons(0, n__zeros))
U511(tt, x0) → ISNATILISTKIND(x0)
U511(tt, n__zeros) → ISNATILISTKIND(n__cons(n__0, n__zeros))
ISNATILISTKIND(n__cons(x0, y1)) → U511(isNatKind(x0), activate(y1))
ISNATILISTKIND(n__cons(n__0, y0)) → U511(isNatKind(n__0), activate(y0))
U511(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(activate(x0), x1))
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
M( U82(x1, ..., x3) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/01\
\00/
·x3

M( U12(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( activate(x1) ) =
/0\
\0/
+
/10\
\01/
·x1

M( n__nil ) =
/0\
\0/

M( n__s(x1) ) =
/0\
\1/
+
/00\
\00/
·x1

M( U21(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( isNatIListKind(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( U51(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( U23(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( tt ) =
/0\
\0/

M( U93(x1, ..., x3) ) =
/0\
\1/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/00\
\00/
·x3

M( isNatList(x1) ) =
/0\
\0/
+
/01\
\00/
·x1

M( U52(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( zeros ) =
/0\
\0/

M( U85(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( s(x1) ) =
/0\
\1/
+
/00\
\00/
·x1

M( isNat(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( U71(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( nil ) =
/0\
\0/

M( U61(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( U83(x1, ..., x3) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/01\
\00/
·x3

M( n__length(x1) ) =
/0\
\1/
+
/00\
\00/
·x1

M( U22(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( U81(x1, ..., x3) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/01\
\00/
·x3

M( n__zeros ) =
/0\
\0/

M( n__cons(x1, x2) ) =
/0\
\0/
+
/10\
\00/
·x1+
/11\
\01/
·x2

M( U11(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( isNatKind(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( 0 ) =
/0\
\0/

M( U94(x1, x2) ) =
/0\
\1/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( cons(x1, x2) ) =
/0\
\0/
+
/10\
\00/
·x1+
/11\
\01/
·x2

M( U91(x1, ..., x3) ) =
/0\
\1/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/00\
\00/
·x3

M( n__0 ) =
/0\
\0/

M( U13(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( U86(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( U92(x1, ..., x3) ) =
/0\
\1/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/00\
\00/
·x3

M( U84(x1, ..., x3) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/01\
\00/
·x3

M( length(x1) ) =
/0\
\1/
+
/00\
\00/
·x1

Tuple symbols:
M( U511(x1, x2) ) = 0+
[0,0]
·x1+
[1,1]
·x2

M( ISNATILISTKIND(x1) ) = 0+
[1,0]
·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)
nil → n__nil
activate(n__zeros) → zeros
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
activate(n__nil) → nil
activate(X) → X
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
U94(tt, L) → s(length(activate(L)))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U86(tt) → tt
U85(tt, V2) → U86(isNatList(activate(V2)))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U71(tt) → tt
U61(tt) → tt
U52(tt) → tt
0 → n__0
length(X) → n__length(X)
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ DependencyGraphProof
                  ↳ AND
                    ↳ QDP
                    ↳ QDP
                    ↳ 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
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ QDPOrderProof
                                                                                                    ↳ QDP
                                                                                                      ↳ QDPOrderProof
                                                                                                        ↳ QDP
                                                                                                          ↳ QDPOrderProof
                                                                                                            ↳ QDP
                                                                                                              ↳ QDPOrderProof
                                                                                                                ↳ QDP
                                                                                                                  ↳ QDPOrderProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ NonTerminationProof
                    ↳ QDP
                    ↳ QDP
                    ↳ QDP
          ↳ QDP

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

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

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X


s = ISNATILISTKIND(n__cons(n__0, n__zeros)) evaluates to t =ISNATILISTKIND(n__cons(n__0, n__zeros))

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



Rewriting sequence

ISNATILISTKIND(n__cons(n__0, n__zeros)) → U511(isNatKind(n__0), activate(n__zeros))
with rule ISNATILISTKIND(n__cons(x0, y1)) → U511(isNatKind(x0), activate(y1)) at position [] and matcher [x0 / n__0, y1 / n__zeros]

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

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

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

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


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





↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ DependencyGraphProof
                  ↳ AND
                    ↳ QDP
                    ↳ QDP
                    ↳ QDP
                    ↳ QDP
                    ↳ QDP
                    ↳ QDP
          ↳ QDP

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

U211(tt, V1) → U221(isNatKind(activate(V1)), activate(V1))
U221(tt, V1) → ISNAT(activate(V1))
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(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ DependencyGraphProof
                  ↳ AND
                    ↳ QDP
                    ↳ QDP
                    ↳ QDP
                    ↳ QDP
                    ↳ QDP
                    ↳ QDP
          ↳ QDP

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

U851(tt, V2) → ISNATLIST(activate(V2))
U821(tt, V1, V2) → U831(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U811(tt, V1, V2) → U821(isNatKind(activate(V1)), activate(V1), activate(V2))
ISNATLIST(n__cons(V1, V2)) → U811(isNatKind(activate(V1)), activate(V1), activate(V2))
U841(tt, V1, V2) → U851(isNat(activate(V1)), activate(V2))
U831(tt, V1, V2) → U841(isNatIListKind(activate(V2)), activate(V1), activate(V2))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ DependencyGraphProof
                  ↳ AND
                    ↳ QDP
                    ↳ QDP
                    ↳ QDP
                    ↳ QDP
                    ↳ QDP
                    ↳ QDP
          ↳ QDP

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

U941(tt, L) → LENGTH(activate(L))
U931(tt, L, N) → U941(isNatKind(activate(N)), activate(L))
LENGTH(cons(N, L)) → U911(isNatList(activate(L)), activate(L), N)
U911(tt, L, N) → U921(isNatIListKind(activate(L)), activate(L), activate(N))
U921(tt, L, N) → U931(isNat(activate(N)), activate(L), activate(N))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ Narrowing

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

U431(tt, V1, V2) → U441(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U421(tt, V1, V2) → U431(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U411(tt, V1, V2) → U421(isNatKind(activate(V1)), activate(V1), activate(V2))
U441(tt, V1, V2) → U451(isNat(activate(V1)), activate(V2))
ISNATILIST(n__cons(V1, V2)) → U411(isNatKind(activate(V1)), activate(V1), activate(V2))
U451(tt, V2) → ISNATILIST(activate(V2))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U441(tt, V1, V2) → U451(isNat(activate(V1)), activate(V2)) at position [0] we obtained the following new rules:

U441(tt, x0, y1) → U451(isNat(x0), activate(y1))
U441(tt, n__length(x0), y1) → U451(isNat(length(activate(x0))), activate(y1))
U441(tt, n__0, y1) → U451(isNat(0), activate(y1))
U441(tt, n__cons(x0, x1), y1) → U451(isNat(cons(activate(x0), x1)), activate(y1))
U441(tt, n__zeros, y1) → U451(isNat(zeros), activate(y1))
U441(tt, n__nil, y1) → U451(isNat(nil), activate(y1))
U441(tt, n__s(x0), y1) → U451(isNat(s(activate(x0))), activate(y1))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ Narrowing
              ↳ QDP
                ↳ Narrowing

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

U431(tt, V1, V2) → U441(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U421(tt, V1, V2) → U431(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U441(tt, x0, y1) → U451(isNat(x0), activate(y1))
U441(tt, n__length(x0), y1) → U451(isNat(length(activate(x0))), activate(y1))
U441(tt, n__0, y1) → U451(isNat(0), activate(y1))
U451(tt, V2) → ISNATILIST(activate(V2))
U411(tt, V1, V2) → U421(isNatKind(activate(V1)), activate(V1), activate(V2))
ISNATILIST(n__cons(V1, V2)) → U411(isNatKind(activate(V1)), activate(V1), activate(V2))
U441(tt, n__cons(x0, x1), y1) → U451(isNat(cons(activate(x0), x1)), activate(y1))
U441(tt, n__zeros, y1) → U451(isNat(zeros), activate(y1))
U441(tt, n__nil, y1) → U451(isNat(nil), activate(y1))
U441(tt, n__s(x0), y1) → U451(isNat(s(activate(x0))), activate(y1))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U451(tt, V2) → ISNATILIST(activate(V2)) at position [0] we obtained the following new rules:

U451(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
U451(tt, n__0) → ISNATILIST(0)
U451(tt, n__zeros) → ISNATILIST(zeros)
U451(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
U451(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
U451(tt, n__nil) → ISNATILIST(nil)
U451(tt, x0) → ISNATILIST(x0)



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ Narrowing
              ↳ QDP
                ↳ Narrowing
                  ↳ QDP
                    ↳ Narrowing

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

U431(tt, V1, V2) → U441(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U451(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
U441(tt, x0, y1) → U451(isNat(x0), activate(y1))
U421(tt, V1, V2) → U431(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U441(tt, n__length(x0), y1) → U451(isNat(length(activate(x0))), activate(y1))
U451(tt, n__zeros) → ISNATILIST(zeros)
U441(tt, n__0, y1) → U451(isNat(0), activate(y1))
U451(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
U411(tt, V1, V2) → U421(isNatKind(activate(V1)), activate(V1), activate(V2))
U451(tt, n__0) → ISNATILIST(0)
U441(tt, n__cons(x0, x1), y1) → U451(isNat(cons(activate(x0), x1)), activate(y1))
ISNATILIST(n__cons(V1, V2)) → U411(isNatKind(activate(V1)), activate(V1), activate(V2))
U441(tt, n__zeros, y1) → U451(isNat(zeros), activate(y1))
U451(tt, n__nil) → ISNATILIST(nil)
U451(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
U441(tt, n__nil, y1) → U451(isNat(nil), activate(y1))
U451(tt, x0) → ISNATILIST(x0)
U441(tt, n__s(x0), y1) → U451(isNat(s(activate(x0))), activate(y1))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U441(tt, n__0, y1) → U451(isNat(0), activate(y1)) at position [0] we obtained the following new rules:

U441(tt, n__0, y0) → U451(isNat(n__0), activate(y0))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ Narrowing
              ↳ QDP
                ↳ Narrowing
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ Narrowing

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

U431(tt, V1, V2) → U441(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U421(tt, V1, V2) → U431(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U441(tt, x0, y1) → U451(isNat(x0), activate(y1))
U451(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
U441(tt, n__length(x0), y1) → U451(isNat(length(activate(x0))), activate(y1))
U451(tt, n__zeros) → ISNATILIST(zeros)
U451(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
U411(tt, V1, V2) → U421(isNatKind(activate(V1)), activate(V1), activate(V2))
U451(tt, n__0) → ISNATILIST(0)
ISNATILIST(n__cons(V1, V2)) → U411(isNatKind(activate(V1)), activate(V1), activate(V2))
U441(tt, n__cons(x0, x1), y1) → U451(isNat(cons(activate(x0), x1)), activate(y1))
U441(tt, n__0, y0) → U451(isNat(n__0), activate(y0))
U441(tt, n__zeros, y1) → U451(isNat(zeros), activate(y1))
U451(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
U451(tt, n__nil) → ISNATILIST(nil)
U441(tt, n__nil, y1) → U451(isNat(nil), activate(y1))
U451(tt, x0) → ISNATILIST(x0)
U441(tt, n__s(x0), y1) → U451(isNat(s(activate(x0))), activate(y1))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U441(tt, n__zeros, y1) → U451(isNat(zeros), activate(y1)) at position [0] we obtained the following new rules:

U441(tt, n__zeros, y0) → U451(isNat(cons(0, n__zeros)), activate(y0))
U441(tt, n__zeros, y0) → U451(isNat(n__zeros), activate(y0))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ Narrowing
              ↳ QDP
                ↳ Narrowing
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof

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

U431(tt, V1, V2) → U441(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U451(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
U441(tt, x0, y1) → U451(isNat(x0), activate(y1))
U421(tt, V1, V2) → U431(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U441(tt, n__length(x0), y1) → U451(isNat(length(activate(x0))), activate(y1))
U451(tt, n__zeros) → ISNATILIST(zeros)
U441(tt, n__zeros, y0) → U451(isNat(n__zeros), activate(y0))
U441(tt, n__zeros, y0) → U451(isNat(cons(0, n__zeros)), activate(y0))
U451(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
U411(tt, V1, V2) → U421(isNatKind(activate(V1)), activate(V1), activate(V2))
U451(tt, n__0) → ISNATILIST(0)
U441(tt, n__cons(x0, x1), y1) → U451(isNat(cons(activate(x0), x1)), activate(y1))
ISNATILIST(n__cons(V1, V2)) → U411(isNatKind(activate(V1)), activate(V1), activate(V2))
U441(tt, n__0, y0) → U451(isNat(n__0), activate(y0))
U451(tt, n__nil) → ISNATILIST(nil)
U451(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
U441(tt, n__nil, y1) → U451(isNat(nil), activate(y1))
U451(tt, x0) → ISNATILIST(x0)
U441(tt, n__s(x0), y1) → U451(isNat(s(activate(x0))), activate(y1))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ Narrowing
              ↳ QDP
                ↳ Narrowing
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Narrowing

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

U431(tt, V1, V2) → U441(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U421(tt, V1, V2) → U431(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U441(tt, x0, y1) → U451(isNat(x0), activate(y1))
U451(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
U441(tt, n__length(x0), y1) → U451(isNat(length(activate(x0))), activate(y1))
U451(tt, n__zeros) → ISNATILIST(zeros)
U441(tt, n__zeros, y0) → U451(isNat(cons(0, n__zeros)), activate(y0))
U451(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
U411(tt, V1, V2) → U421(isNatKind(activate(V1)), activate(V1), activate(V2))
U451(tt, n__0) → ISNATILIST(0)
U441(tt, n__cons(x0, x1), y1) → U451(isNat(cons(activate(x0), x1)), activate(y1))
ISNATILIST(n__cons(V1, V2)) → U411(isNatKind(activate(V1)), activate(V1), activate(V2))
U441(tt, n__0, y0) → U451(isNat(n__0), activate(y0))
U451(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
U451(tt, n__nil) → ISNATILIST(nil)
U441(tt, n__nil, y1) → U451(isNat(nil), activate(y1))
U451(tt, x0) → ISNATILIST(x0)
U441(tt, n__s(x0), y1) → U451(isNat(s(activate(x0))), activate(y1))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U451(tt, n__0) → ISNATILIST(0) at position [0] we obtained the following new rules:

U451(tt, n__0) → ISNATILIST(n__0)



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ Narrowing
              ↳ QDP
                ↳ Narrowing
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ DependencyGraphProof

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

U431(tt, V1, V2) → U441(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U451(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
U441(tt, x0, y1) → U451(isNat(x0), activate(y1))
U421(tt, V1, V2) → U431(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U441(tt, n__length(x0), y1) → U451(isNat(length(activate(x0))), activate(y1))
U451(tt, n__zeros) → ISNATILIST(zeros)
U451(tt, n__0) → ISNATILIST(n__0)
U441(tt, n__zeros, y0) → U451(isNat(cons(0, n__zeros)), activate(y0))
U451(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
U411(tt, V1, V2) → U421(isNatKind(activate(V1)), activate(V1), activate(V2))
ISNATILIST(n__cons(V1, V2)) → U411(isNatKind(activate(V1)), activate(V1), activate(V2))
U441(tt, n__cons(x0, x1), y1) → U451(isNat(cons(activate(x0), x1)), activate(y1))
U441(tt, n__0, y0) → U451(isNat(n__0), activate(y0))
U451(tt, n__nil) → ISNATILIST(nil)
U451(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
U441(tt, n__nil, y1) → U451(isNat(nil), activate(y1))
U451(tt, x0) → ISNATILIST(x0)
U441(tt, n__s(x0), y1) → U451(isNat(s(activate(x0))), activate(y1))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ Narrowing
              ↳ QDP
                ↳ Narrowing
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Narrowing

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

U431(tt, V1, V2) → U441(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U421(tt, V1, V2) → U431(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U441(tt, x0, y1) → U451(isNat(x0), activate(y1))
U451(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
U441(tt, n__length(x0), y1) → U451(isNat(length(activate(x0))), activate(y1))
U451(tt, n__zeros) → ISNATILIST(zeros)
U441(tt, n__zeros, y0) → U451(isNat(cons(0, n__zeros)), activate(y0))
U451(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
U411(tt, V1, V2) → U421(isNatKind(activate(V1)), activate(V1), activate(V2))
ISNATILIST(n__cons(V1, V2)) → U411(isNatKind(activate(V1)), activate(V1), activate(V2))
U441(tt, n__cons(x0, x1), y1) → U451(isNat(cons(activate(x0), x1)), activate(y1))
U441(tt, n__0, y0) → U451(isNat(n__0), activate(y0))
U451(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
U451(tt, n__nil) → ISNATILIST(nil)
U441(tt, n__nil, y1) → U451(isNat(nil), activate(y1))
U451(tt, x0) → ISNATILIST(x0)
U441(tt, n__s(x0), y1) → U451(isNat(s(activate(x0))), activate(y1))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U451(tt, n__zeros) → ISNATILIST(zeros) at position [0] we obtained the following new rules:

U451(tt, n__zeros) → ISNATILIST(n__zeros)
U451(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ Narrowing
              ↳ QDP
                ↳ Narrowing
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ QDP
                                            ↳ DependencyGraphProof

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

U431(tt, V1, V2) → U441(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U451(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
U441(tt, x0, y1) → U451(isNat(x0), activate(y1))
U421(tt, V1, V2) → U431(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U441(tt, n__length(x0), y1) → U451(isNat(length(activate(x0))), activate(y1))
U441(tt, n__zeros, y0) → U451(isNat(cons(0, n__zeros)), activate(y0))
U451(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
U411(tt, V1, V2) → U421(isNatKind(activate(V1)), activate(V1), activate(V2))
U441(tt, n__cons(x0, x1), y1) → U451(isNat(cons(activate(x0), x1)), activate(y1))
ISNATILIST(n__cons(V1, V2)) → U411(isNatKind(activate(V1)), activate(V1), activate(V2))
U441(tt, n__0, y0) → U451(isNat(n__0), activate(y0))
U451(tt, n__zeros) → ISNATILIST(n__zeros)
U451(tt, n__nil) → ISNATILIST(nil)
U451(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
U441(tt, n__nil, y1) → U451(isNat(nil), activate(y1))
U451(tt, x0) → ISNATILIST(x0)
U451(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
U441(tt, n__s(x0), y1) → U451(isNat(s(activate(x0))), activate(y1))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ Narrowing
              ↳ QDP
                ↳ Narrowing
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Narrowing

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

U431(tt, V1, V2) → U441(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U441(tt, x0, y1) → U451(isNat(x0), activate(y1))
U451(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
U421(tt, V1, V2) → U431(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U441(tt, n__length(x0), y1) → U451(isNat(length(activate(x0))), activate(y1))
U441(tt, n__zeros, y0) → U451(isNat(cons(0, n__zeros)), activate(y0))
U451(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
U411(tt, V1, V2) → U421(isNatKind(activate(V1)), activate(V1), activate(V2))
ISNATILIST(n__cons(V1, V2)) → U411(isNatKind(activate(V1)), activate(V1), activate(V2))
U441(tt, n__cons(x0, x1), y1) → U451(isNat(cons(activate(x0), x1)), activate(y1))
U441(tt, n__0, y0) → U451(isNat(n__0), activate(y0))
U451(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
U451(tt, n__nil) → ISNATILIST(nil)
U441(tt, n__nil, y1) → U451(isNat(nil), activate(y1))
U451(tt, x0) → ISNATILIST(x0)
U441(tt, n__s(x0), y1) → U451(isNat(s(activate(x0))), activate(y1))
U451(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U451(tt, n__nil) → ISNATILIST(nil) at position [0] we obtained the following new rules:

U451(tt, n__nil) → ISNATILIST(n__nil)



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ Narrowing
              ↳ QDP
                ↳ Narrowing
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof

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

U431(tt, V1, V2) → U441(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U451(tt, n__nil) → ISNATILIST(n__nil)
U421(tt, V1, V2) → U431(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U451(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
U441(tt, x0, y1) → U451(isNat(x0), activate(y1))
U441(tt, n__length(x0), y1) → U451(isNat(length(activate(x0))), activate(y1))
U441(tt, n__zeros, y0) → U451(isNat(cons(0, n__zeros)), activate(y0))
U451(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
U411(tt, V1, V2) → U421(isNatKind(activate(V1)), activate(V1), activate(V2))
U441(tt, n__cons(x0, x1), y1) → U451(isNat(cons(activate(x0), x1)), activate(y1))
ISNATILIST(n__cons(V1, V2)) → U411(isNatKind(activate(V1)), activate(V1), activate(V2))
U441(tt, n__0, y0) → U451(isNat(n__0), activate(y0))
U451(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
U441(tt, n__nil, y1) → U451(isNat(nil), activate(y1))
U451(tt, x0) → ISNATILIST(x0)
U451(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
U441(tt, n__s(x0), y1) → U451(isNat(s(activate(x0))), activate(y1))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ Narrowing
              ↳ QDP
                ↳ Narrowing
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Narrowing

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

U431(tt, V1, V2) → U441(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U441(tt, x0, y1) → U451(isNat(x0), activate(y1))
U451(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
U421(tt, V1, V2) → U431(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U441(tt, n__length(x0), y1) → U451(isNat(length(activate(x0))), activate(y1))
U441(tt, n__zeros, y0) → U451(isNat(cons(0, n__zeros)), activate(y0))
U451(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
U411(tt, V1, V2) → U421(isNatKind(activate(V1)), activate(V1), activate(V2))
ISNATILIST(n__cons(V1, V2)) → U411(isNatKind(activate(V1)), activate(V1), activate(V2))
U441(tt, n__cons(x0, x1), y1) → U451(isNat(cons(activate(x0), x1)), activate(y1))
U441(tt, n__0, y0) → U451(isNat(n__0), activate(y0))
U451(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
U441(tt, n__nil, y1) → U451(isNat(nil), activate(y1))
U451(tt, x0) → ISNATILIST(x0)
U441(tt, n__s(x0), y1) → U451(isNat(s(activate(x0))), activate(y1))
U451(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U451(tt, n__zeros) → ISNATILIST(cons(0, n__zeros)) at position [0] we obtained the following new rules:

U451(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U451(tt, n__zeros) → ISNATILIST(cons(n__0, n__zeros))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ Narrowing
              ↳ QDP
                ↳ Narrowing
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Narrowing
                                                          ↳ QDP
                                                            ↳ Narrowing

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

U431(tt, V1, V2) → U441(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U421(tt, V1, V2) → U431(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U451(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
U441(tt, x0, y1) → U451(isNat(x0), activate(y1))
U451(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U441(tt, n__length(x0), y1) → U451(isNat(length(activate(x0))), activate(y1))
U441(tt, n__zeros, y0) → U451(isNat(cons(0, n__zeros)), activate(y0))
U451(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
U451(tt, n__zeros) → ISNATILIST(cons(n__0, n__zeros))
U411(tt, V1, V2) → U421(isNatKind(activate(V1)), activate(V1), activate(V2))
U441(tt, n__cons(x0, x1), y1) → U451(isNat(cons(activate(x0), x1)), activate(y1))
ISNATILIST(n__cons(V1, V2)) → U411(isNatKind(activate(V1)), activate(V1), activate(V2))
U441(tt, n__0, y0) → U451(isNat(n__0), activate(y0))
U451(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
U441(tt, n__nil, y1) → U451(isNat(nil), activate(y1))
U451(tt, x0) → ISNATILIST(x0)
U441(tt, n__s(x0), y1) → U451(isNat(s(activate(x0))), activate(y1))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U441(tt, n__nil, y1) → U451(isNat(nil), activate(y1)) at position [0] we obtained the following new rules:

U441(tt, n__nil, y0) → U451(isNat(n__nil), activate(y0))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ Narrowing
              ↳ QDP
                ↳ Narrowing
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Narrowing
                                                          ↳ QDP
                                                            ↳ Narrowing
                                                              ↳ QDP
                                                                ↳ DependencyGraphProof

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

U431(tt, V1, V2) → U441(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U441(tt, x0, y1) → U451(isNat(x0), activate(y1))
U451(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
U421(tt, V1, V2) → U431(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U441(tt, n__length(x0), y1) → U451(isNat(length(activate(x0))), activate(y1))
U451(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U441(tt, n__zeros, y0) → U451(isNat(cons(0, n__zeros)), activate(y0))
U451(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
U441(tt, n__nil, y0) → U451(isNat(n__nil), activate(y0))
U451(tt, n__zeros) → ISNATILIST(cons(n__0, n__zeros))
U411(tt, V1, V2) → U421(isNatKind(activate(V1)), activate(V1), activate(V2))
ISNATILIST(n__cons(V1, V2)) → U411(isNatKind(activate(V1)), activate(V1), activate(V2))
U441(tt, n__cons(x0, x1), y1) → U451(isNat(cons(activate(x0), x1)), activate(y1))
U441(tt, n__0, y0) → U451(isNat(n__0), activate(y0))
U451(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
U451(tt, x0) → ISNATILIST(x0)
U441(tt, n__s(x0), y1) → U451(isNat(s(activate(x0))), activate(y1))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ Narrowing
              ↳ QDP
                ↳ Narrowing
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Narrowing
                                                          ↳ QDP
                                                            ↳ Narrowing
                                                              ↳ QDP
                                                                ↳ DependencyGraphProof
                                                                  ↳ QDP
                                                                    ↳ Narrowing

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

U431(tt, V1, V2) → U441(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U451(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
U421(tt, V1, V2) → U431(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U441(tt, x0, y1) → U451(isNat(x0), activate(y1))
U451(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U441(tt, n__length(x0), y1) → U451(isNat(length(activate(x0))), activate(y1))
U441(tt, n__zeros, y0) → U451(isNat(cons(0, n__zeros)), activate(y0))
U451(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
U451(tt, n__zeros) → ISNATILIST(cons(n__0, n__zeros))
U411(tt, V1, V2) → U421(isNatKind(activate(V1)), activate(V1), activate(V2))
ISNATILIST(n__cons(V1, V2)) → U411(isNatKind(activate(V1)), activate(V1), activate(V2))
U441(tt, n__cons(x0, x1), y1) → U451(isNat(cons(activate(x0), x1)), activate(y1))
U441(tt, n__0, y0) → U451(isNat(n__0), activate(y0))
U451(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
U451(tt, x0) → ISNATILIST(x0)
U441(tt, n__s(x0), y1) → U451(isNat(s(activate(x0))), activate(y1))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U451(tt, n__zeros) → ISNATILIST(cons(n__0, n__zeros)) at position [0] we obtained the following new rules:

U451(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ Narrowing
              ↳ QDP
                ↳ Narrowing
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Narrowing
                                                          ↳ QDP
                                                            ↳ Narrowing
                                                              ↳ QDP
                                                                ↳ DependencyGraphProof
                                                                  ↳ QDP
                                                                    ↳ Narrowing
                                                                      ↳ QDP
                                                                        ↳ Narrowing

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

U431(tt, V1, V2) → U441(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U441(tt, x0, y1) → U451(isNat(x0), activate(y1))
U421(tt, V1, V2) → U431(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U451(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
U441(tt, n__length(x0), y1) → U451(isNat(length(activate(x0))), activate(y1))
U451(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U451(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))
U441(tt, n__zeros, y0) → U451(isNat(cons(0, n__zeros)), activate(y0))
U451(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
U411(tt, V1, V2) → U421(isNatKind(activate(V1)), activate(V1), activate(V2))
U441(tt, n__cons(x0, x1), y1) → U451(isNat(cons(activate(x0), x1)), activate(y1))
ISNATILIST(n__cons(V1, V2)) → U411(isNatKind(activate(V1)), activate(V1), activate(V2))
U441(tt, n__0, y0) → U451(isNat(n__0), activate(y0))
U451(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
U451(tt, x0) → ISNATILIST(x0)
U441(tt, n__s(x0), y1) → U451(isNat(s(activate(x0))), activate(y1))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U441(tt, n__zeros, y0) → U451(isNat(cons(0, n__zeros)), activate(y0)) at position [0] we obtained the following new rules:

U441(tt, n__zeros, y0) → U451(isNat(n__cons(0, n__zeros)), activate(y0))
U441(tt, n__zeros, y0) → U451(isNat(cons(n__0, n__zeros)), activate(y0))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ Narrowing
              ↳ QDP
                ↳ Narrowing
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Narrowing
                                                          ↳ QDP
                                                            ↳ Narrowing
                                                              ↳ QDP
                                                                ↳ DependencyGraphProof
                                                                  ↳ QDP
                                                                    ↳ Narrowing
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof

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

U441(tt, n__zeros, y0) → U451(isNat(n__cons(0, n__zeros)), activate(y0))
U431(tt, V1, V2) → U441(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U451(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
U421(tt, V1, V2) → U431(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U441(tt, x0, y1) → U451(isNat(x0), activate(y1))
U451(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U441(tt, n__length(x0), y1) → U451(isNat(length(activate(x0))), activate(y1))
U451(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))
U451(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
U411(tt, V1, V2) → U421(isNatKind(activate(V1)), activate(V1), activate(V2))
U441(tt, n__zeros, y0) → U451(isNat(cons(n__0, n__zeros)), activate(y0))
ISNATILIST(n__cons(V1, V2)) → U411(isNatKind(activate(V1)), activate(V1), activate(V2))
U441(tt, n__cons(x0, x1), y1) → U451(isNat(cons(activate(x0), x1)), activate(y1))
U441(tt, n__0, y0) → U451(isNat(n__0), activate(y0))
U451(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
U451(tt, x0) → ISNATILIST(x0)
U441(tt, n__s(x0), y1) → U451(isNat(s(activate(x0))), activate(y1))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ Narrowing
              ↳ QDP
                ↳ Narrowing
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Narrowing
                                                          ↳ QDP
                                                            ↳ Narrowing
                                                              ↳ QDP
                                                                ↳ DependencyGraphProof
                                                                  ↳ QDP
                                                                    ↳ Narrowing
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Narrowing

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

U431(tt, V1, V2) → U441(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U421(tt, V1, V2) → U431(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U441(tt, x0, y1) → U451(isNat(x0), activate(y1))
U451(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
U451(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U441(tt, n__length(x0), y1) → U451(isNat(length(activate(x0))), activate(y1))
U451(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))
U451(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
U411(tt, V1, V2) → U421(isNatKind(activate(V1)), activate(V1), activate(V2))
U441(tt, n__zeros, y0) → U451(isNat(cons(n__0, n__zeros)), activate(y0))
ISNATILIST(n__cons(V1, V2)) → U411(isNatKind(activate(V1)), activate(V1), activate(V2))
U441(tt, n__cons(x0, x1), y1) → U451(isNat(cons(activate(x0), x1)), activate(y1))
U441(tt, n__0, y0) → U451(isNat(n__0), activate(y0))
U451(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
U451(tt, x0) → ISNATILIST(x0)
U441(tt, n__s(x0), y1) → U451(isNat(s(activate(x0))), activate(y1))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U441(tt, n__zeros, y0) → U451(isNat(cons(n__0, n__zeros)), activate(y0)) at position [0] we obtained the following new rules:

U441(tt, n__zeros, y0) → U451(isNat(n__cons(n__0, n__zeros)), activate(y0))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ Narrowing
              ↳ QDP
                ↳ Narrowing
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Narrowing
                                                          ↳ QDP
                                                            ↳ Narrowing
                                                              ↳ QDP
                                                                ↳ DependencyGraphProof
                                                                  ↳ QDP
                                                                    ↳ Narrowing
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Narrowing
                                                                                  ↳ QDP
                                                                                    ↳ DependencyGraphProof

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

U431(tt, V1, V2) → U441(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U451(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
U441(tt, x0, y1) → U451(isNat(x0), activate(y1))
U421(tt, V1, V2) → U431(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U441(tt, n__length(x0), y1) → U451(isNat(length(activate(x0))), activate(y1))
U451(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U451(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))
U451(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
U411(tt, V1, V2) → U421(isNatKind(activate(V1)), activate(V1), activate(V2))
U441(tt, n__zeros, y0) → U451(isNat(n__cons(n__0, n__zeros)), activate(y0))
U441(tt, n__cons(x0, x1), y1) → U451(isNat(cons(activate(x0), x1)), activate(y1))
ISNATILIST(n__cons(V1, V2)) → U411(isNatKind(activate(V1)), activate(V1), activate(V2))
U441(tt, n__0, y0) → U451(isNat(n__0), activate(y0))
U451(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
U451(tt, x0) → ISNATILIST(x0)
U441(tt, n__s(x0), y1) → U451(isNat(s(activate(x0))), activate(y1))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ Narrowing
              ↳ QDP
                ↳ Narrowing
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Narrowing
                                                          ↳ QDP
                                                            ↳ Narrowing
                                                              ↳ QDP
                                                                ↳ DependencyGraphProof
                                                                  ↳ QDP
                                                                    ↳ Narrowing
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Narrowing
                                                                                  ↳ QDP
                                                                                    ↳ DependencyGraphProof
                                                                                      ↳ QDP
                                                                                        ↳ QDPOrderProof

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

U431(tt, V1, V2) → U441(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U421(tt, V1, V2) → U431(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U441(tt, x0, y1) → U451(isNat(x0), activate(y1))
U451(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
U451(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U441(tt, n__length(x0), y1) → U451(isNat(length(activate(x0))), activate(y1))
U451(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))
U451(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
U411(tt, V1, V2) → U421(isNatKind(activate(V1)), activate(V1), activate(V2))
ISNATILIST(n__cons(V1, V2)) → U411(isNatKind(activate(V1)), activate(V1), activate(V2))
U441(tt, n__cons(x0, x1), y1) → U451(isNat(cons(activate(x0), x1)), activate(y1))
U441(tt, n__0, y0) → U451(isNat(n__0), activate(y0))
U451(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
U451(tt, x0) → ISNATILIST(x0)
U441(tt, n__s(x0), y1) → U451(isNat(s(activate(x0))), activate(y1))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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


The following pairs can be oriented strictly and are deleted.


U441(tt, n__length(x0), y1) → U451(isNat(length(activate(x0))), activate(y1))
The remaining pairs can at least be oriented weakly.

U431(tt, V1, V2) → U441(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U421(tt, V1, V2) → U431(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U441(tt, x0, y1) → U451(isNat(x0), activate(y1))
U451(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
U451(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U451(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))
U451(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
U411(tt, V1, V2) → U421(isNatKind(activate(V1)), activate(V1), activate(V2))
ISNATILIST(n__cons(V1, V2)) → U411(isNatKind(activate(V1)), activate(V1), activate(V2))
U441(tt, n__cons(x0, x1), y1) → U451(isNat(cons(activate(x0), x1)), activate(y1))
U441(tt, n__0, y0) → U451(isNat(n__0), activate(y0))
U451(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
U451(tt, x0) → ISNATILIST(x0)
U441(tt, n__s(x0), y1) → U451(isNat(s(activate(x0))), activate(y1))
Used ordering: Polynomial interpretation [25]:

POL(0) = 0   
POL(ISNATILIST(x1)) = x1   
POL(U11(x1, x2)) = 0   
POL(U12(x1, x2)) = 0   
POL(U13(x1)) = 0   
POL(U21(x1, x2)) = 0   
POL(U22(x1, x2)) = 0   
POL(U23(x1)) = 0   
POL(U411(x1, x2, x3)) = x2 + x3   
POL(U421(x1, x2, x3)) = x2 + x3   
POL(U431(x1, x2, x3)) = x2 + x3   
POL(U441(x1, x2, x3)) = x2 + x3   
POL(U451(x1, x2)) = x2   
POL(U51(x1, x2)) = 0   
POL(U52(x1)) = 0   
POL(U61(x1)) = 0   
POL(U71(x1)) = 0   
POL(U81(x1, x2, x3)) = x2 + x3   
POL(U82(x1, x2, x3)) = 0   
POL(U83(x1, x2, x3)) = 0   
POL(U84(x1, x2, x3)) = 0   
POL(U85(x1, x2)) = 0   
POL(U86(x1)) = 0   
POL(U91(x1, x2, x3)) = 0   
POL(U92(x1, x2, x3)) = 0   
POL(U93(x1, x2, x3)) = 0   
POL(U94(x1, x2)) = 0   
POL(activate(x1)) = x1   
POL(cons(x1, x2)) = x1 + x2   
POL(isNat(x1)) = 0   
POL(isNatIListKind(x1)) = 0   
POL(isNatKind(x1)) = 0   
POL(isNatList(x1)) = x1   
POL(length(x1)) = 1   
POL(n__0) = 0   
POL(n__cons(x1, x2)) = x1 + x2   
POL(n__length(x1)) = 1   
POL(n__nil) = 0   
POL(n__s(x1)) = 0   
POL(n__zeros) = 0   
POL(nil) = 0   
POL(s(x1)) = 0   
POL(tt) = 0   
POL(zeros) = 0   

The following usable rules [17] were oriented:

zeros → cons(0, n__zeros)
nil → n__nil
activate(n__zeros) → zeros
s(X) → n__s(X)
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
cons(X1, X2) → n__cons(X1, X2)
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
activate(n__nil) → nil
activate(X) → X
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
U94(tt, L) → s(length(activate(L)))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U86(tt) → tt
U85(tt, V2) → U86(isNatList(activate(V2)))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U71(tt) → tt
U61(tt) → tt
U52(tt) → tt
0 → n__0
length(X) → n__length(X)
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ Narrowing
              ↳ QDP
                ↳ Narrowing
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Narrowing
                                                          ↳ QDP
                                                            ↳ Narrowing
                                                              ↳ QDP
                                                                ↳ DependencyGraphProof
                                                                  ↳ QDP
                                                                    ↳ Narrowing
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Narrowing
                                                                                  ↳ QDP
                                                                                    ↳ DependencyGraphProof
                                                                                      ↳ QDP
                                                                                        ↳ QDPOrderProof
                                                                                          ↳ QDP
                                                                                            ↳ QDPOrderProof

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

U431(tt, V1, V2) → U441(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U421(tt, V1, V2) → U431(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U441(tt, x0, y1) → U451(isNat(x0), activate(y1))
U451(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
U451(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U451(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))
U451(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
U411(tt, V1, V2) → U421(isNatKind(activate(V1)), activate(V1), activate(V2))
ISNATILIST(n__cons(V1, V2)) → U411(isNatKind(activate(V1)), activate(V1), activate(V2))
U441(tt, n__cons(x0, x1), y1) → U451(isNat(cons(activate(x0), x1)), activate(y1))
U441(tt, n__0, y0) → U451(isNat(n__0), activate(y0))
U451(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
U451(tt, x0) → ISNATILIST(x0)
U441(tt, n__s(x0), y1) → U451(isNat(s(activate(x0))), activate(y1))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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


The following pairs can be oriented strictly and are deleted.


U441(tt, n__s(x0), y1) → U451(isNat(s(activate(x0))), activate(y1))
The remaining pairs can at least be oriented weakly.

U431(tt, V1, V2) → U441(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U421(tt, V1, V2) → U431(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U441(tt, x0, y1) → U451(isNat(x0), activate(y1))
U451(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
U451(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U451(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))
U451(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
U411(tt, V1, V2) → U421(isNatKind(activate(V1)), activate(V1), activate(V2))
ISNATILIST(n__cons(V1, V2)) → U411(isNatKind(activate(V1)), activate(V1), activate(V2))
U441(tt, n__cons(x0, x1), y1) → U451(isNat(cons(activate(x0), x1)), activate(y1))
U441(tt, n__0, y0) → U451(isNat(n__0), activate(y0))
U451(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
U451(tt, x0) → ISNATILIST(x0)
Used ordering: Polynomial interpretation [25]:

POL(0) = 0   
POL(ISNATILIST(x1)) = x1   
POL(U11(x1, x2)) = 0   
POL(U12(x1, x2)) = 0   
POL(U13(x1)) = 0   
POL(U21(x1, x2)) = 0   
POL(U22(x1, x2)) = 0   
POL(U23(x1)) = 0   
POL(U411(x1, x2, x3)) = x2 + x3   
POL(U421(x1, x2, x3)) = x2 + x3   
POL(U431(x1, x2, x3)) = x2 + x3   
POL(U441(x1, x2, x3)) = x2 + x3   
POL(U451(x1, x2)) = x2   
POL(U51(x1, x2)) = 0   
POL(U52(x1)) = 0   
POL(U61(x1)) = 0   
POL(U71(x1)) = 0   
POL(U81(x1, x2, x3)) = x2   
POL(U82(x1, x2, x3)) = x2   
POL(U83(x1, x2, x3)) = 0   
POL(U84(x1, x2, x3)) = 0   
POL(U85(x1, x2)) = 0   
POL(U86(x1)) = 0   
POL(U91(x1, x2, x3)) = 1   
POL(U92(x1, x2, x3)) = 1   
POL(U93(x1, x2, x3)) = 1   
POL(U94(x1, x2)) = 1   
POL(activate(x1)) = x1   
POL(cons(x1, x2)) = x1 + x2   
POL(isNat(x1)) = 0   
POL(isNatIListKind(x1)) = 0   
POL(isNatKind(x1)) = 0   
POL(isNatList(x1)) = x1   
POL(length(x1)) = 1   
POL(n__0) = 0   
POL(n__cons(x1, x2)) = x1 + x2   
POL(n__length(x1)) = 1   
POL(n__nil) = 0   
POL(n__s(x1)) = 1   
POL(n__zeros) = 0   
POL(nil) = 0   
POL(s(x1)) = 1   
POL(tt) = 0   
POL(zeros) = 0   

The following usable rules [17] were oriented:

zeros → cons(0, n__zeros)
nil → n__nil
activate(n__zeros) → zeros
s(X) → n__s(X)
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
cons(X1, X2) → n__cons(X1, X2)
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
activate(n__nil) → nil
activate(X) → X
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
U94(tt, L) → s(length(activate(L)))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U86(tt) → tt
U85(tt, V2) → U86(isNatList(activate(V2)))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U71(tt) → tt
U61(tt) → tt
U52(tt) → tt
0 → n__0
length(X) → n__length(X)
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ Narrowing
              ↳ QDP
                ↳ Narrowing
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Narrowing
                                                          ↳ QDP
                                                            ↳ Narrowing
                                                              ↳ QDP
                                                                ↳ DependencyGraphProof
                                                                  ↳ QDP
                                                                    ↳ Narrowing
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Narrowing
                                                                                  ↳ QDP
                                                                                    ↳ DependencyGraphProof
                                                                                      ↳ QDP
                                                                                        ↳ QDPOrderProof
                                                                                          ↳ QDP
                                                                                            ↳ QDPOrderProof
                                                                                              ↳ QDP
                                                                                                ↳ QDPOrderProof

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

U431(tt, V1, V2) → U441(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U421(tt, V1, V2) → U431(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U441(tt, x0, y1) → U451(isNat(x0), activate(y1))
U451(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
U451(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U451(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))
U451(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
U411(tt, V1, V2) → U421(isNatKind(activate(V1)), activate(V1), activate(V2))
ISNATILIST(n__cons(V1, V2)) → U411(isNatKind(activate(V1)), activate(V1), activate(V2))
U441(tt, n__cons(x0, x1), y1) → U451(isNat(cons(activate(x0), x1)), activate(y1))
U441(tt, n__0, y0) → U451(isNat(n__0), activate(y0))
U451(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
U451(tt, x0) → ISNATILIST(x0)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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


The following pairs can be oriented strictly and are deleted.


U441(tt, n__cons(x0, x1), y1) → U451(isNat(cons(activate(x0), x1)), activate(y1))
The remaining pairs can at least be oriented weakly.

U431(tt, V1, V2) → U441(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U421(tt, V1, V2) → U431(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U441(tt, x0, y1) → U451(isNat(x0), activate(y1))
U451(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
U451(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U451(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))
U451(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
U411(tt, V1, V2) → U421(isNatKind(activate(V1)), activate(V1), activate(V2))
ISNATILIST(n__cons(V1, V2)) → U411(isNatKind(activate(V1)), activate(V1), activate(V2))
U441(tt, n__0, y0) → U451(isNat(n__0), activate(y0))
U451(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
U451(tt, x0) → ISNATILIST(x0)
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
M( U82(x1, ..., x3) ) =
/1\
\1/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/00\
\00/
·x3

M( U12(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( activate(x1) ) =
/0\
\0/
+
/10\
\01/
·x1

M( n__nil ) =
/0\
\0/

M( n__s(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( U21(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( U51(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( isNatIListKind(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( U23(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( tt ) =
/0\
\0/

M( U93(x1, ..., x3) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/00\
\00/
·x3

M( isNatList(x1) ) =
/1\
\0/
+
/11\
\11/
·x1

M( U52(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( zeros ) =
/1\
\0/

M( U85(x1, x2) ) =
/1\
\1/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( s(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( isNat(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( U71(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( nil ) =
/0\
\0/

M( U61(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( U83(x1, ..., x3) ) =
/1\
\1/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/00\
\00/
·x3

M( n__length(x1) ) =
/0\
\0/
+
/01\
\00/
·x1

M( U22(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( U81(x1, ..., x3) ) =
/1\
\1/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/00\
\00/
·x3

M( n__zeros ) =
/1\
\0/

M( n__cons(x1, x2) ) =
/1\
\0/
+
/00\
\10/
·x1+
/00\
\01/
·x2

M( U11(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( isNatKind(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( 0 ) =
/0\
\0/

M( U94(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( cons(x1, x2) ) =
/1\
\0/
+
/00\
\10/
·x1+
/00\
\01/
·x2

M( U91(x1, ..., x3) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/00\
\00/
·x3

M( n__0 ) =
/0\
\0/

M( U13(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( U86(x1) ) =
/1\
\1/
+
/00\
\00/
·x1

M( U92(x1, ..., x3) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/00\
\00/
·x3

M( U84(x1, ..., x3) ) =
/1\
\1/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/00\
\00/
·x3

M( length(x1) ) =
/0\
\0/
+
/01\
\00/
·x1

Tuple symbols:
M( U431(x1, ..., x3) ) = 0+
[0,0]
·x1+
[1,0]
·x2+
[0,1]
·x3

M( U421(x1, ..., x3) ) = 0+
[0,0]
·x1+
[1,0]
·x2+
[0,1]
·x3

M( U451(x1, x2) ) = 0+
[0,0]
·x1+
[0,1]
·x2

M( ISNATILIST(x1) ) = 0+
[0,1]
·x1

M( U411(x1, ..., x3) ) = 0+
[0,0]
·x1+
[1,0]
·x2+
[0,1]
·x3

M( U441(x1, ..., x3) ) = 0+
[0,0]
·x1+
[1,0]
·x2+
[0,1]
·x3


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)
nil → n__nil
activate(n__zeros) → zeros
s(X) → n__s(X)
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
cons(X1, X2) → n__cons(X1, X2)
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
activate(n__nil) → nil
activate(X) → X
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
U94(tt, L) → s(length(activate(L)))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U86(tt) → tt
U85(tt, V2) → U86(isNatList(activate(V2)))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U71(tt) → tt
U61(tt) → tt
U52(tt) → tt
0 → n__0
length(X) → n__length(X)
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ Narrowing
              ↳ QDP
                ↳ Narrowing
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ QDP
                                                        ↳ Narrowing
                                                          ↳ QDP
                                                            ↳ Narrowing
                                                              ↳ QDP
                                                                ↳ DependencyGraphProof
                                                                  ↳ QDP
                                                                    ↳ Narrowing
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Narrowing
                                                                                  ↳ QDP
                                                                                    ↳ DependencyGraphProof
                                                                                      ↳ QDP
                                                                                        ↳ QDPOrderProof
                                                                                          ↳ QDP
                                                                                            ↳ QDPOrderProof
                                                                                              ↳ QDP
                                                                                                ↳ QDPOrderProof
                                                                                                  ↳ QDP
                                                                                                    ↳ QDPOrderProof

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

U431(tt, V1, V2) → U441(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U411(tt, V1, V2) → U421(isNatKind(activate(V1)), activate(V1), activate(V2))
U451(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
U441(tt, x0, y1) → U451(isNat(x0), activate(y1))
U421(tt, V1, V2) → U431(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U451(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U451(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))
ISNATILIST(n__cons(V1, V2)) → U411(isNatKind(activate(V1)), activate(V1), activate(V2))
U441(tt, n__0, y0) → U451(isNat(n__0), activate(y0))
U451(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
U451(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
U451(tt, x0) → ISNATILIST(x0)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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


The following pairs can be oriented strictly and are deleted.


U451(tt, n__length(x0)) → ISNATILIST(length(activate(x0)))
The remaining pairs can at least be oriented weakly.

U431(tt, V1, V2) → U441(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U411(tt, V1, V2) → U421(isNatKind(activate(V1)), activate(V1), activate(V2))
U441(tt, x0, y1) → U451(isNat(x0), activate(y1))
U421(tt, V1, V2) → U431(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U451(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U451(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))
ISNATILIST(n__cons(V1, V2)) → U411(isNatKind(activate(V1)), activate(V1), activate(V2))
U441(tt, n__0, y0) → U451(isNat(n__0), activate(y0))
U451(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
U451(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
U451(tt, x0) → ISNATILIST(x0)
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
M( U82(x1, ..., x3) ) =
/1\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/11\
\00/
·x3

M( U12(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( activate(x1) ) =
/0\
\0/
+
/10\
\01/
·x1

M( n__nil ) =
/0\
\0/

M( n__s(x1) ) =
/0\
\0/
+
/00\
\01/
·x1

M( U21(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( U51(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( isNatIListKind(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( U23(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( tt ) =
/0\
\0/

M( U93(x1, ..., x3) ) =
/0\
\1/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/00\
\00/
·x3

M( isNatList(x1) ) =
/1\
\0/
+
/10\
\00/
·x1

M( U52(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( zeros ) =
/0\
\0/

M( U85(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( s(x1) ) =
/0\
\0/
+
/00\
\01/
·x1

M( isNat(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( U71(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( nil ) =
/0\
\0/

M( U61(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( U83(x1, ..., x3) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/10\
\00/
·x3

M( n__length(x1) ) =
/0\
\1/
+
/00\
\00/
·x1

M( U22(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( U81(x1, ..., x3) ) =
/1\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/11\
\00/
·x3

M( n__zeros ) =
/0\
\0/

M( n__cons(x1, x2) ) =
/0\
\0/
+
/10\
\00/
·x1+
/11\
\00/
·x2

M( U11(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( isNatKind(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( 0 ) =
/0\
\0/

M( U94(x1, x2) ) =
/0\
\1/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( cons(x1, x2) ) =
/0\
\0/
+
/10\
\00/
·x1+
/11\
\00/
·x2

M( U91(x1, ..., x3) ) =
/0\
\1/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/00\
\00/
·x3

M( n__0 ) =
/0\
\0/

M( U13(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( U86(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( U92(x1, ..., x3) ) =
/0\
\1/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/00\
\00/
·x3

M( U84(x1, ..., x3) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/10\
\00/
·x3

M( length(x1) ) =
/0\
\1/
+
/00\
\00/
·x1

Tuple symbols:
M( U431(x1, ..., x3) ) = 0+
[0,0]
·x1+
[0,0]
·x2+
[1,1]
·x3

M( U421(x1, ..., x3) ) = 0+
[0,0]
·x1+
[0,0]
·x2+
[1,1]
·x3

M( U451(x1, x2) ) = 0+
[0,0]
·x1+
[1,1]
·x2

M( ISNATILIST(x1) ) = 0+
[1,0]
·x1

M( U411(x1, ..., x3) ) = 0+
[0,0]
·x1+
[0,0]
·x2+
[1,1]
·x3

M( U441(x1, ..., x3) ) = 0+
[0,0]
·x1+
[0,0]
·x2+
[1,1]
·x3


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)
nil → n__nil
activate(n__zeros) → zeros
s(X) → n__s(X)
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
cons(X1, X2) → n__cons(X1, X2)
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
activate(n__nil) → nil
activate(X) → X
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
U94(tt, L) → s(length(activate(L)))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U86(tt) → tt
U85(tt, V2) → U86(isNatList(activate(V2)))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U71(tt) → tt
U61(tt) → tt
U52(tt) → tt
0 → n__0
length(X) → n__length(X)
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ Narrowing
              ↳ QDP
                ↳ Narrowing
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ 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

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

U431(tt, V1, V2) → U441(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U421(tt, V1, V2) → U431(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U441(tt, x0, y1) → U451(isNat(x0), activate(y1))
U411(tt, V1, V2) → U421(isNatKind(activate(V1)), activate(V1), activate(V2))
U451(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U451(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))
ISNATILIST(n__cons(V1, V2)) → U411(isNatKind(activate(V1)), activate(V1), activate(V2))
U441(tt, n__0, y0) → U451(isNat(n__0), activate(y0))
U451(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
U451(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
U451(tt, x0) → ISNATILIST(x0)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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


The following pairs can be oriented strictly and are deleted.


U451(tt, n__s(x0)) → ISNATILIST(s(activate(x0)))
The remaining pairs can at least be oriented weakly.

U431(tt, V1, V2) → U441(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U421(tt, V1, V2) → U431(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U441(tt, x0, y1) → U451(isNat(x0), activate(y1))
U411(tt, V1, V2) → U421(isNatKind(activate(V1)), activate(V1), activate(V2))
U451(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U451(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))
ISNATILIST(n__cons(V1, V2)) → U411(isNatKind(activate(V1)), activate(V1), activate(V2))
U441(tt, n__0, y0) → U451(isNat(n__0), activate(y0))
U451(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
U451(tt, x0) → ISNATILIST(x0)
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
M( U82(x1, ..., x3) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/00\
\00/
·x3

M( U12(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( activate(x1) ) =
/0\
\0/
+
/10\
\01/
·x1

M( n__nil ) =
/0\
\0/

M( n__s(x1) ) =
/0\
\1/
+
/00\
\00/
·x1

M( U21(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( U51(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( isNatIListKind(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( U23(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( tt ) =
/0\
\0/

M( U93(x1, ..., x3) ) =
/0\
\1/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/00\
\00/
·x3

M( isNatList(x1) ) =
/0\
\0/
+
/11\
\01/
·x1

M( U52(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( zeros ) =
/0\
\0/

M( U85(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( s(x1) ) =
/0\
\1/
+
/00\
\00/
·x1

M( isNat(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( U71(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( nil ) =
/0\
\0/

M( U61(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( U83(x1, ..., x3) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/00\
\00/
·x3

M( n__length(x1) ) =
/0\
\1/
+
/00\
\00/
·x1

M( U22(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( U81(x1, ..., x3) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\01/
·x2+
/00\
\00/
·x3

M( n__zeros ) =
/0\
\0/

M( n__cons(x1, x2) ) =
/0\
\0/
+
/00\
\01/
·x1+
/11\
\00/
·x2

M( U11(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( isNatKind(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( 0 ) =
/0\
\0/

M( U94(x1, x2) ) =
/0\
\1/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( cons(x1, x2) ) =
/0\
\0/
+
/00\
\01/
·x1+
/11\
\00/
·x2

M( U91(x1, ..., x3) ) =
/0\
\1/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/00\
\00/
·x3

M( n__0 ) =
/0\
\0/

M( U13(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( U86(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( U92(x1, ..., x3) ) =
/0\
\1/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/00\
\00/
·x3

M( U84(x1, ..., x3) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2+
/00\
\00/
·x3

M( length(x1) ) =
/0\
\1/
+
/00\
\00/
·x1

Tuple symbols:
M( U431(x1, ..., x3) ) = 0+
[0,0]
·x1+
[0,0]
·x2+
[1,1]
·x3

M( U421(x1, ..., x3) ) = 0+
[0,0]
·x1+
[0,0]
·x2+
[1,1]
·x3

M( U451(x1, x2) ) = 0+
[0,0]
·x1+
[1,1]
·x2

M( ISNATILIST(x1) ) = 0+
[1,0]
·x1

M( U411(x1, ..., x3) ) = 0+
[0,0]
·x1+
[0,0]
·x2+
[1,1]
·x3

M( U441(x1, ..., x3) ) = 0+
[0,0]
·x1+
[0,0]
·x2+
[1,1]
·x3


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)
nil → n__nil
activate(n__zeros) → zeros
s(X) → n__s(X)
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
cons(X1, X2) → n__cons(X1, X2)
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
activate(n__nil) → nil
activate(X) → X
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
U94(tt, L) → s(length(activate(L)))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U86(tt) → tt
U85(tt, V2) → U86(isNatList(activate(V2)))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U71(tt) → tt
U61(tt) → tt
U52(tt) → tt
0 → n__0
length(X) → n__length(X)
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ Narrowing
              ↳ QDP
                ↳ Narrowing
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ DependencyGraphProof
                                                      ↳ 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

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

U431(tt, V1, V2) → U441(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U411(tt, V1, V2) → U421(isNatKind(activate(V1)), activate(V1), activate(V2))
U441(tt, x0, y1) → U451(isNat(x0), activate(y1))
U421(tt, V1, V2) → U431(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U451(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U451(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))
ISNATILIST(n__cons(V1, V2)) → U411(isNatKind(activate(V1)), activate(V1), activate(V2))
U441(tt, n__0, y0) → U451(isNat(n__0), activate(y0))
U451(tt, n__cons(x0, x1)) → ISNATILIST(cons(activate(x0), x1))
U451(tt, x0) → ISNATILIST(x0)

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
U12(tt, V1) → U13(isNatList(activate(V1)))
U13(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V) → U32(isNatIListKind(activate(V)), activate(V))
U32(tt, V) → U33(isNatList(activate(V)))
U33(tt) → tt
U41(tt, V1, V2) → U42(isNatKind(activate(V1)), activate(V1), activate(V2))
U42(tt, V1, V2) → U43(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U43(tt, V1, V2) → U44(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U44(tt, V1, V2) → U45(isNat(activate(V1)), activate(V2))
U45(tt, V2) → U46(isNatIList(activate(V2)))
U46(tt) → tt
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U81(tt, V1, V2) → U82(isNatKind(activate(V1)), activate(V1), activate(V2))
U82(tt, V1, V2) → U83(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U83(tt, V1, V2) → U84(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U84(tt, V1, V2) → U85(isNat(activate(V1)), activate(V2))
U85(tt, V2) → U86(isNatList(activate(V2)))
U86(tt) → tt
U91(tt, L, N) → U92(isNatIListKind(activate(L)), activate(L), activate(N))
U92(tt, L, N) → U93(isNat(activate(N)), activate(L), activate(N))
U93(tt, L, N) → U94(isNatKind(activate(N)), activate(L))
U94(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → U51(isNatKind(activate(V1)), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(activate(V1)))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__nil) → nil
activate(X) → X

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