Termination w.r.t. Q proof of /tmp/tmpcAhyKi/LengthOfFiniteLists_complete-noand

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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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))
ACTIVATE(n__length(X)) → LENGTH(X)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
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))
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)
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)
ACTIVATE(n__cons(X1, X2)) → CONS(X1, X2)
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)
ACTIVATE(n__s(X)) → S(X)
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)
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)

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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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))
ACTIVATE(n__length(X)) → LENGTH(X)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
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))
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)
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)
ACTIVATE(n__cons(X1, X2)) → CONS(X1, X2)
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)
ACTIVATE(n__s(X)) → S(X)
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)
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)

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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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))
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)
ACTIVATE(n__length(X)) → LENGTH(X)
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)
U12¹(tt, V1) → ISNATLIST(activate(V1))
U83¹(tt, V1, V2) → ISNATILISTKIND(activate(V2))
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))
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)
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)
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)
ISNAT(n__length(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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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)
ISNAT(n__length(V1)) → U11¹(isNatIListKind(activate(V1)), activate(V1))
ISNATKIND(n__length(V1)) → ISNATILISTKIND(activate(V1))
ACTIVATE(n__length(X)) → LENGTH(X)
ISNAT(n__length(V1)) → ISNATILISTKIND(activate(V1))
ISNAT(n__length(V1)) → ACTIVATE(V1)

The remaining pairs can at least be oriented weakly.

U92¹(tt, L, N) → ACTIVATE(L)
U81¹(tt, V1, V2) → ISNATKIND(activate(V1))
U21¹(tt, V1) → ISNATKIND(activate(V1))
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)
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)
U12¹(tt, V1) → ISNATLIST(activate(V1))
U83¹(tt, V1, V2) → ISNATILISTKIND(activate(V2))
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))
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)
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)
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)

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)) = 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)) = 0
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 + x₃
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)) = 0
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) = 0
POL(n__s(x₁)) = x₁
POL(n__zeros) = 0
POL(nil) = 0
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(X) → X
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
U11(tt, V1) → U12(isNatIListKind(activate(V1)), activate(V1))
zeros → cons(0, n__zeros)
U13(tt) → tt
U12(tt, V1) → U13(isNatList(activate(V1)))
U51(tt, V2) → U52(isNatIListKind(activate(V2)))
U52(tt) → tt
U61(tt) → tt
U71(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
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))
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
length(X) → n__length(X)
s(X) → n__s(X)
zeros → n__zeros
0 → n__0
length(nil) → 0
length(cons(N, L)) → U91(isNatList(activate(L)), activate(L), N)
isNatList(n__cons(V1, V2)) → U81(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatKind(n__length(V1)) → U61(isNatIListKind(activate(V1)))
isNatKind(n__s(V1)) → U71(isNatKind(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))
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)
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)
U12¹(tt, V1) → ISNATLIST(activate(V1))
U83¹(tt, V1, V2) → ISNATILISTKIND(activate(V2))
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))
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))
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))
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)

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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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 5 SCCs with 43 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ 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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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: Polynomial interpretation [25,35]:

POL(U82(x₁, x₂, x₃)) = (2)x_3
POL(activate(x₁)) = x_1
POL(U12(x₁, x₂)) = x_2
POL(n__nil) = 1
POL(n__s(x₁)) = 2 + (2)x_1
POL(U21(x₁, x₂)) = 1 + x_2
POL(U51(x₁, x₂)) = 1
POL(isNatIListKind(x₁)) = 2
POL(U23(x₁)) = 1
POL(tt) = 1
POL(U93(x₁, x₂, x₃)) = 2 + (2)x_2
POL(isNatList(x₁)) = x_1
POL(zeros) = 0
POL(U52(x₁)) = 1
POL(U85(x₁, x₂)) = x_2
POL(s(x₁)) = 2 + (2)x_1
POL(isNat(x₁)) = x_1
POL(U71(x₁)) = 1
POL(nil) = 1
POL(U83(x₁, x₂, x₃)) = (2)x_3
POL(U61(x₁)) = 1
POL(n__length(x₁)) = x_1
POL(U22(x₁, x₂)) = 1 + x_2
POL(ISNATKIND(x₁)) = (2)x_1
POL(U81(x₁, x₂, x₃)) = (4)x_3
POL(n__zeros) = 0
POL(n__cons(x₁, x₂)) = (4)x_2
POL(U11(x₁, x₂)) = x_2
POL(isNatKind(x₁)) = 2
POL(0) = 0
POL(U94(x₁, x₂)) = 2 + (2)x_2
POL(cons(x₁, x₂)) = (4)x_2
POL(U91(x₁, x₂, x₃)) = (2)x_1 + (2)x_2
POL(n__0) = 0
POL(U13(x₁)) = x_1
POL(U86(x₁)) = x_1
POL(U92(x₁, x₂, x₃)) = 2 + (2)x_2
POL(U84(x₁, x₂, x₃)) = x_3
POL(length(x₁)) = x_1

The value of delta used in the strict ordering is 4.
The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ 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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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

                      ↳ Narrowing

                    ↳ QDP

                    ↳ QDP

                    ↳ QDP

          ↳ QDP

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

U51¹(tt, V2) → ISNATILISTKIND(activate(V2))
ISNATILISTKIND(n__cons(V1, V2)) → U51¹(isNatKind(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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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(x0)), activate(y1))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → U51¹(isNatKind(length(x0)), 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__cons(x0, x1), y1)) → U51¹(isNatKind(cons(x0, x1)), activate(y1))
ISNATILISTKIND(n__cons(x0, y1)) → U51¹(isNatKind(x0), activate(y1))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ 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(x0)), activate(y1))
ISNATILISTKIND(n__cons(n__0, y1)) → U51¹(isNatKind(0), activate(y1))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → U51¹(isNatKind(length(x0)), activate(y1))
ISNATILISTKIND(n__cons(n__nil, y1)) → U51¹(isNatKind(nil), activate(y1))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U51¹(isNatKind(cons(x0, x1)), activate(y1))
ISNATILISTKIND(n__cons(n__zeros, y1)) → U51¹(isNatKind(zeros), activate(y1))
U51¹(tt, V2) → ISNATILISTKIND(activate(V2))
ISNATILISTKIND(n__cons(x0, y1)) → U51¹(isNatKind(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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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__0) → ISNATILISTKIND(0)
U51¹(tt, n__s(x0)) → ISNATILISTKIND(s(x0))
U51¹(tt, x0) → ISNATILISTKIND(x0)
U51¹(tt, n__nil) → ISNATILISTKIND(nil)
U51¹(tt, n__zeros) → ISNATILISTKIND(zeros)
U51¹(tt, n__length(x0)) → ISNATILISTKIND(length(x0))
U51¹(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(x0, x1))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ 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__length(x0), y1)) → U51¹(isNatKind(length(x0)), activate(y1))
U51¹(tt, n__0) → ISNATILISTKIND(0)
U51¹(tt, n__s(x0)) → ISNATILISTKIND(s(x0))
U51¹(tt, n__nil) → ISNATILISTKIND(nil)
ISNATILISTKIND(n__cons(x0, y1)) → U51¹(isNatKind(x0), activate(y1))
U51¹(tt, n__zeros) → ISNATILISTKIND(zeros)
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U51¹(isNatKind(s(x0)), activate(y1))
ISNATILISTKIND(n__cons(n__nil, y1)) → U51¹(isNatKind(nil), activate(y1))
U51¹(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U51¹(isNatKind(cons(x0, x1)), activate(y1))
ISNATILISTKIND(n__cons(n__zeros, y1)) → U51¹(isNatKind(zeros), activate(y1))
U51¹(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(x0, x1))
U51¹(tt, n__length(x0)) → ISNATILISTKIND(length(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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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

                      ↳ Narrowing

                        ↳ 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__length(x0), y1)) → U51¹(isNatKind(length(x0)), activate(y1))
U51¹(tt, n__0) → ISNATILISTKIND(0)
U51¹(tt, n__s(x0)) → ISNATILISTKIND(s(x0))
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)
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U51¹(isNatKind(s(x0)), activate(y1))
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__cons(x0, x1), y1)) → U51¹(isNatKind(cons(x0, x1)), activate(y1))
U51¹(tt, n__length(x0)) → ISNATILISTKIND(length(x0))
U51¹(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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__s(x0), y1)) → U51¹(isNatKind(s(x0)), activate(y1)) at position [0] we obtained the following new rules:

ISNATILISTKIND(n__cons(n__s(x0), y1)) → U51¹(isNatKind(n__s(x0)), activate(y1))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ 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__length(x0), y1)) → U51¹(isNatKind(length(x0)), activate(y1))
U51¹(tt, n__0) → ISNATILISTKIND(0)
U51¹(tt, n__s(x0)) → ISNATILISTKIND(s(x0))
U51¹(tt, n__nil) → ISNATILISTKIND(nil)
ISNATILISTKIND(n__cons(x0, y1)) → U51¹(isNatKind(x0), activate(y1))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U51¹(isNatKind(n__s(x0)), activate(y1))
ISNATILISTKIND(n__cons(n__0, y0)) → U51¹(isNatKind(n__0), activate(y0))
U51¹(tt, n__zeros) → ISNATILISTKIND(zeros)
ISNATILISTKIND(n__cons(n__nil, y1)) → U51¹(isNatKind(nil), activate(y1))
U51¹(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U51¹(isNatKind(cons(x0, x1)), activate(y1))
ISNATILISTKIND(n__cons(n__zeros, y1)) → U51¹(isNatKind(zeros), activate(y1))
U51¹(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(x0, x1))
U51¹(tt, n__length(x0)) → ISNATILISTKIND(length(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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                    ↳ QDP

                    ↳ QDP

                    ↳ QDP

          ↳ QDP

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

ISNATILISTKIND(n__cons(n__length(x0), y1)) → U51¹(isNatKind(length(x0)), activate(y1))
U51¹(tt, n__0) → ISNATILISTKIND(0)
U51¹(tt, n__s(x0)) → ISNATILISTKIND(s(x0))
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))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U51¹(isNatKind(n__s(x0)), activate(y1))
U51¹(tt, n__zeros) → ISNATILISTKIND(zeros)
U51¹(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(n__zeros, y1)) → U51¹(isNatKind(zeros), activate(y1))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U51¹(isNatKind(cons(x0, x1)), activate(y1))
ISNATILISTKIND(n__cons(n__nil, y0)) → U51¹(isNatKind(n__nil), activate(y0))
U51¹(tt, n__length(x0)) → ISNATILISTKIND(length(x0))
U51¹(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                    ↳ QDP

                    ↳ QDP

                    ↳ QDP

          ↳ QDP

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

ISNATILISTKIND(n__cons(n__length(x0), y1)) → U51¹(isNatKind(length(x0)), activate(y1))
U51¹(tt, n__0) → ISNATILISTKIND(0)
U51¹(tt, n__s(x0)) → ISNATILISTKIND(s(x0))
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))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U51¹(isNatKind(n__s(x0)), activate(y1))
U51¹(tt, n__zeros) → ISNATILISTKIND(zeros)
U51¹(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(n__zeros, y1)) → U51¹(isNatKind(zeros), activate(y1))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U51¹(isNatKind(cons(x0, x1)), activate(y1))
U51¹(tt, n__length(x0)) → ISNATILISTKIND(length(x0))
U51¹(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ Narrowing

                                ↳ 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__length(x0), y1)) → U51¹(isNatKind(length(x0)), activate(y1))
U51¹(tt, n__s(x0)) → ISNATILISTKIND(s(x0))
U51¹(tt, n__nil) → ISNATILISTKIND(nil)
ISNATILISTKIND(n__cons(x0, y1)) → U51¹(isNatKind(x0), activate(y1))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U51¹(isNatKind(n__s(x0)), activate(y1))
ISNATILISTKIND(n__cons(n__0, y0)) → U51¹(isNatKind(n__0), activate(y0))
U51¹(tt, n__zeros) → ISNATILISTKIND(zeros)
U51¹(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U51¹(isNatKind(cons(x0, x1)), activate(y1))
ISNATILISTKIND(n__cons(n__zeros, y1)) → U51¹(isNatKind(zeros), activate(y1))
U51¹(tt, n__0) → ISNATILISTKIND(n__0)
U51¹(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(x0, x1))
U51¹(tt, n__length(x0)) → ISNATILISTKIND(length(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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ Narrowing

                                ↳ 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:

ISNATILISTKIND(n__cons(n__length(x0), y1)) → U51¹(isNatKind(length(x0)), activate(y1))
U51¹(tt, x0) → ISNATILISTKIND(x0)
U51¹(tt, n__s(x0)) → ISNATILISTKIND(s(x0))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U51¹(isNatKind(cons(x0, x1)), activate(y1))
ISNATILISTKIND(n__cons(n__zeros, y1)) → U51¹(isNatKind(zeros), activate(y1))
U51¹(tt, n__nil) → ISNATILISTKIND(nil)
ISNATILISTKIND(n__cons(x0, y1)) → U51¹(isNatKind(x0), activate(y1))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U51¹(isNatKind(n__s(x0)), activate(y1))
ISNATILISTKIND(n__cons(n__0, y0)) → U51¹(isNatKind(n__0), activate(y0))
U51¹(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(x0, x1))
U51¹(tt, n__length(x0)) → ISNATILISTKIND(length(x0))
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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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__s(x0)) → ISNATILISTKIND(s(x0)) at position [0] we obtained the following new rules:

U51¹(tt, n__s(x0)) → ISNATILISTKIND(n__s(x0))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ Narrowing

                                ↳ 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__length(x0), y1)) → U51¹(isNatKind(length(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))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U51¹(isNatKind(n__s(x0)), activate(y1))
U51¹(tt, n__zeros) → ISNATILISTKIND(zeros)
U51¹(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(n__zeros, y1)) → U51¹(isNatKind(zeros), activate(y1))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U51¹(isNatKind(cons(x0, x1)), activate(y1))
U51¹(tt, n__s(x0)) → ISNATILISTKIND(n__s(x0))
U51¹(tt, n__length(x0)) → ISNATILISTKIND(length(x0))
U51¹(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ Narrowing

                                ↳ 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:

ISNATILISTKIND(n__cons(n__length(x0), y1)) → U51¹(isNatKind(length(x0)), activate(y1))
U51¹(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U51¹(isNatKind(cons(x0, x1)), activate(y1))
ISNATILISTKIND(n__cons(n__zeros, y1)) → U51¹(isNatKind(zeros), activate(y1))
U51¹(tt, n__nil) → ISNATILISTKIND(nil)
ISNATILISTKIND(n__cons(x0, y1)) → U51¹(isNatKind(x0), activate(y1))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U51¹(isNatKind(n__s(x0)), activate(y1))
ISNATILISTKIND(n__cons(n__0, y0)) → U51¹(isNatKind(n__0), activate(y0))
U51¹(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(x0, x1))
U51¹(tt, n__length(x0)) → ISNATILISTKIND(length(x0))
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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ Narrowing

                                ↳ 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__length(x0), y1)) → U51¹(isNatKind(length(x0)), activate(y1))
U51¹(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(n__zeros, y1)) → U51¹(isNatKind(zeros), activate(y1))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U51¹(isNatKind(cons(x0, x1)), activate(y1))
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__s(x0), y1)) → U51¹(isNatKind(n__s(x0)), activate(y1))
U51¹(tt, n__nil) → ISNATILISTKIND(n__nil)
U51¹(tt, n__zeros) → ISNATILISTKIND(zeros)
U51¹(tt, n__length(x0)) → ISNATILISTKIND(length(x0))
U51¹(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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

                      ↳ Narrowing

                        ↳ 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

                    ↳ QDP

                    ↳ QDP

          ↳ QDP

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

ISNATILISTKIND(n__cons(n__length(x0), y1)) → U51¹(isNatKind(length(x0)), activate(y1))
U51¹(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U51¹(isNatKind(cons(x0, x1)), activate(y1))
ISNATILISTKIND(n__cons(n__zeros, y1)) → U51¹(isNatKind(zeros), activate(y1))
ISNATILISTKIND(n__cons(x0, y1)) → U51¹(isNatKind(x0), activate(y1))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U51¹(isNatKind(n__s(x0)), activate(y1))
ISNATILISTKIND(n__cons(n__0, y0)) → U51¹(isNatKind(n__0), activate(y0))
U51¹(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(x0, x1))
U51¹(tt, n__length(x0)) → ISNATILISTKIND(length(x0))
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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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

                      ↳ Narrowing

                        ↳ 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

                                                                          ↳ DependencyGraphProof

                    ↳ QDP

                    ↳ QDP

                    ↳ QDP

          ↳ QDP

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

ISNATILISTKIND(n__cons(n__length(x0), y1)) → U51¹(isNatKind(length(x0)), activate(y1))
ISNATILISTKIND(n__cons(n__zeros, y0)) → U51¹(isNatKind(cons(0, n__zeros)), activate(y0))
U51¹(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U51¹(isNatKind(cons(x0, x1)), activate(y1))
ISNATILISTKIND(n__cons(n__zeros, y0)) → U51¹(isNatKind(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__s(x0), y1)) → U51¹(isNatKind(n__s(x0)), activate(y1))
U51¹(tt, n__zeros) → ISNATILISTKIND(zeros)
U51¹(tt, n__length(x0)) → ISNATILISTKIND(length(x0))
U51¹(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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

                      ↳ Narrowing

                        ↳ 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

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                    ↳ QDP

                    ↳ QDP

                    ↳ QDP

          ↳ QDP

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

ISNATILISTKIND(n__cons(n__length(x0), y1)) → U51¹(isNatKind(length(x0)), activate(y1))
ISNATILISTKIND(n__cons(n__zeros, y0)) → U51¹(isNatKind(cons(0, n__zeros)), activate(y0))
U51¹(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U51¹(isNatKind(cons(x0, x1)), activate(y1))
ISNATILISTKIND(n__cons(x0, y1)) → U51¹(isNatKind(x0), activate(y1))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U51¹(isNatKind(n__s(x0)), activate(y1))
ISNATILISTKIND(n__cons(n__0, y0)) → U51¹(isNatKind(n__0), activate(y0))
U51¹(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(x0, x1))
U51¹(tt, n__length(x0)) → ISNATILISTKIND(length(x0))
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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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

                      ↳ Narrowing

                        ↳ 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

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                    ↳ QDP

                    ↳ QDP

                    ↳ QDP

          ↳ QDP

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

ISNATILISTKIND(n__cons(n__length(x0), y1)) → U51¹(isNatKind(length(x0)), activate(y1))
U51¹(tt, n__zeros) → ISNATILISTKIND(cons(0, n__zeros))
U51¹(tt, n__zeros) → ISNATILISTKIND(n__zeros)
ISNATILISTKIND(n__cons(n__zeros, y0)) → U51¹(isNatKind(cons(0, n__zeros)), activate(y0))
U51¹(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U51¹(isNatKind(cons(x0, x1)), activate(y1))
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__s(x0), y1)) → U51¹(isNatKind(n__s(x0)), activate(y1))
U51¹(tt, n__length(x0)) → ISNATILISTKIND(length(x0))
U51¹(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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

                      ↳ Narrowing

                        ↳ 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

                                                                          ↳ DependencyGraphProof

                                                                            ↳ 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__length(x0), y1)) → U51¹(isNatKind(length(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))
U51¹(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → U51¹(isNatKind(cons(x0, x1)), activate(y1))
ISNATILISTKIND(n__cons(x0, y1)) → U51¹(isNatKind(x0), activate(y1))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U51¹(isNatKind(n__s(x0)), activate(y1))
ISNATILISTKIND(n__cons(n__0, y0)) → U51¹(isNatKind(n__0), activate(y0))
U51¹(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(x0, x1))
U51¹(tt, n__length(x0)) → ISNATILISTKIND(length(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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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__cons(x0, x1), y1)) → U51¹(isNatKind(cons(x0, x1)), activate(y1)) at position [0] we obtained the following new rules:

ISNATILISTKIND(n__cons(n__cons(x0, x1), y2)) → U51¹(isNatKind(n__cons(x0, x1)), activate(y2))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ Narrowing

                        ↳ 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

                                                                          ↳ 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__length(x0), y1)) → U51¹(isNatKind(length(x0)), activate(y1))
U51¹(tt, n__zeros) → ISNATILISTKIND(cons(0, n__zeros))
ISNATILISTKIND(n__cons(n__zeros, y0)) → U51¹(isNatKind(cons(0, n__zeros)), activate(y0))
U51¹(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(n__cons(x0, x1), y2)) → U51¹(isNatKind(n__cons(x0, x1)), activate(y2))
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__s(x0), y1)) → U51¹(isNatKind(n__s(x0)), activate(y1))
U51¹(tt, n__length(x0)) → ISNATILISTKIND(length(x0))
U51¹(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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

                      ↳ Narrowing

                        ↳ 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

                                                                          ↳ 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:

ISNATILISTKIND(n__cons(n__length(x0), y1)) → U51¹(isNatKind(length(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))
U51¹(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(x0, y1)) → U51¹(isNatKind(x0), activate(y1))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U51¹(isNatKind(n__s(x0)), activate(y1))
ISNATILISTKIND(n__cons(n__0, y0)) → U51¹(isNatKind(n__0), activate(y0))
U51¹(tt, n__cons(x0, x1)) → ISNATILISTKIND(cons(x0, x1))
U51¹(tt, n__length(x0)) → ISNATILISTKIND(length(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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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__cons(x0, x1)) → ISNATILISTKIND(cons(x0, x1)) at position [0] we obtained the following new rules:

U51¹(tt, n__cons(x0, x1)) → ISNATILISTKIND(n__cons(x0, x1))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ Narrowing

                        ↳ 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

                                                                          ↳ 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__length(x0), y1)) → U51¹(isNatKind(length(x0)), activate(y1))
U51¹(tt, n__zeros) → ISNATILISTKIND(cons(0, n__zeros))
ISNATILISTKIND(n__cons(n__zeros, y0)) → U51¹(isNatKind(cons(0, n__zeros)), activate(y0))
U51¹(tt, x0) → ISNATILISTKIND(x0)
U51¹(tt, n__cons(x0, x1)) → ISNATILISTKIND(n__cons(x0, x1))
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__s(x0), y1)) → U51¹(isNatKind(n__s(x0)), activate(y1))
U51¹(tt, n__length(x0)) → ISNATILISTKIND(length(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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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

                      ↳ Narrowing

                        ↳ 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

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                    ↳ QDP

                    ↳ QDP

                    ↳ QDP

          ↳ QDP

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

U51¹(tt, n__zeros) → ISNATILISTKIND(n__cons(0, n__zeros))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → U51¹(isNatKind(length(x0)), activate(y1))
ISNATILISTKIND(n__cons(n__zeros, y0)) → U51¹(isNatKind(cons(0, n__zeros)), activate(y0))
U51¹(tt, x0) → ISNATILISTKIND(x0)
U51¹(tt, n__cons(x0, x1)) → ISNATILISTKIND(n__cons(x0, x1))
ISNATILISTKIND(n__cons(x0, y1)) → U51¹(isNatKind(x0), activate(y1))
U51¹(tt, n__zeros) → ISNATILISTKIND(cons(n__0, n__zeros))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U51¹(isNatKind(n__s(x0)), activate(y1))
ISNATILISTKIND(n__cons(n__0, y0)) → U51¹(isNatKind(n__0), activate(y0))
U51¹(tt, n__length(x0)) → ISNATILISTKIND(length(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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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

                      ↳ Narrowing

                        ↳ 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

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                    ↳ QDP

                    ↳ QDP

                    ↳ QDP

          ↳ QDP

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

U51¹(tt, n__zeros) → ISNATILISTKIND(n__cons(0, n__zeros))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → U51¹(isNatKind(length(x0)), activate(y1))
U51¹(tt, x0) → ISNATILISTKIND(x0)
ISNATILISTKIND(n__cons(n__zeros, y0)) → U51¹(isNatKind(n__cons(0, n__zeros)), activate(y0))
U51¹(tt, n__cons(x0, x1)) → ISNATILISTKIND(n__cons(x0, x1))
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__s(x0), y1)) → U51¹(isNatKind(n__s(x0)), activate(y1))
U51¹(tt, n__zeros) → ISNATILISTKIND(cons(n__0, n__zeros))
ISNATILISTKIND(n__cons(n__zeros, y0)) → U51¹(isNatKind(cons(n__0, n__zeros)), activate(y0))
U51¹(tt, n__length(x0)) → ISNATILISTKIND(length(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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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

                      ↳ Narrowing

                        ↳ 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

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                    ↳ QDP

                    ↳ QDP

                    ↳ QDP

          ↳ QDP

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

U51¹(tt, n__zeros) → ISNATILISTKIND(n__cons(0, n__zeros))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → U51¹(isNatKind(length(x0)), activate(y1))
U51¹(tt, x0) → ISNATILISTKIND(x0)
U51¹(tt, n__cons(x0, x1)) → ISNATILISTKIND(n__cons(x0, x1))
ISNATILISTKIND(n__cons(x0, y1)) → U51¹(isNatKind(x0), activate(y1))
U51¹(tt, n__zeros) → ISNATILISTKIND(cons(n__0, n__zeros))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U51¹(isNatKind(n__s(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__length(x0)) → ISNATILISTKIND(length(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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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

                      ↳ Narrowing

                        ↳ 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

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                    ↳ QDP

                    ↳ QDP

                    ↳ QDP

          ↳ QDP

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

U51¹(tt, n__zeros) → ISNATILISTKIND(n__cons(0, n__zeros))
ISNATILISTKIND(n__cons(n__zeros, y0)) → U51¹(isNatKind(n__cons(n__0, n__zeros)), activate(y0))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → U51¹(isNatKind(length(x0)), activate(y1))
U51¹(tt, x0) → ISNATILISTKIND(x0)
U51¹(tt, n__cons(x0, x1)) → ISNATILISTKIND(n__cons(x0, x1))
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__s(x0), y1)) → U51¹(isNatKind(n__s(x0)), activate(y1))
U51¹(tt, n__zeros) → ISNATILISTKIND(cons(n__0, n__zeros))
U51¹(tt, n__length(x0)) → ISNATILISTKIND(length(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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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

                      ↳ Narrowing

                        ↳ 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

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ 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))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → U51¹(isNatKind(length(x0)), activate(y1))
U51¹(tt, x0) → ISNATILISTKIND(x0)
U51¹(tt, n__cons(x0, x1)) → ISNATILISTKIND(n__cons(x0, x1))
ISNATILISTKIND(n__cons(x0, y1)) → U51¹(isNatKind(x0), activate(y1))
U51¹(tt, n__zeros) → ISNATILISTKIND(cons(n__0, n__zeros))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U51¹(isNatKind(n__s(x0)), activate(y1))
ISNATILISTKIND(n__cons(n__0, y0)) → U51¹(isNatKind(n__0), activate(y0))
U51¹(tt, n__length(x0)) → ISNATILISTKIND(length(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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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

                      ↳ Narrowing

                        ↳ 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

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ 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__zeros) → ISNATILISTKIND(n__cons(0, n__zeros))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → U51¹(isNatKind(length(x0)), activate(y1))
U51¹(tt, x0) → ISNATILISTKIND(x0)
U51¹(tt, n__cons(x0, x1)) → ISNATILISTKIND(n__cons(x0, x1))
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))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U51¹(isNatKind(n__s(x0)), activate(y1))
U51¹(tt, n__length(x0)) → ISNATILISTKIND(length(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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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(x0)), activate(y1))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → U51¹(isNatKind(n__s(x0)), activate(y1))

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__cons(x0, x1)) → ISNATILISTKIND(n__cons(x0, x1))
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__length(x0)) → ISNATILISTKIND(length(x0))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(ISNATILISTKIND(x₁)) = x₁
POL(U11(x₁, x₂)) = 1 + x₂
POL(U12(x₁, x₂)) = 0
POL(U13(x₁)) = 0
POL(U21(x₁, x₂)) = 1
POL(U22(x₁, x₂)) = 1
POL(U23(x₁)) = 0
POL(U51(x₁, x₂)) = 1
POL(U51¹(x₁, x₂)) = x₂
POL(U52(x₁)) = 1
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)) = 1 + x₂ + x₃
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)) = x₁
POL(isNatIListKind(x₁)) = 1
POL(isNatKind(x₁)) = 0
POL(isNatList(x₁)) = 0
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) = 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:

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

                      ↳ Narrowing

                        ↳ 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

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ 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, x0) → ISNATILISTKIND(x0)
U51¹(tt, n__cons(x0, x1)) → ISNATILISTKIND(n__cons(x0, x1))
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__length(x0)) → ISNATILISTKIND(length(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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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(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__cons(x0, x1)) → ISNATILISTKIND(n__cons(x0, x1))
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))

Used ordering: Polynomial interpretation [25,35]:

POL(U51¹(x₁, x₂)) = (4)x_2
POL(U82(x₁, x₂, x₃)) = 3 + (4)x_2 + x_3
POL(ISNATILISTKIND(x₁)) = x_1
POL(activate(x₁)) = x_1
POL(U12(x₁, x₂)) = 1
POL(n__nil) = 0
POL(n__s(x₁)) = 0
POL(U51(x₁, x₂)) = 4
POL(isNatIListKind(x₁)) = 4
POL(U21(x₁, x₂)) = 2
POL(U23(x₁)) = 0
POL(tt) = 0
POL(U93(x₁, x₂, x₃)) = 0
POL(isNatList(x₁)) = 3 + (2)x_1
POL(zeros) = 0
POL(U52(x₁)) = 0
POL(U85(x₁, x₂)) = 1
POL(s(x₁)) = 0
POL(isNat(x₁)) = 2
POL(U71(x₁)) = 3
POL(nil) = 0
POL(U83(x₁, x₂, x₃)) = 3 + (4)x_2
POL(U61(x₁)) = 3
POL(n__length(x₁)) = 2
POL(U22(x₁, x₂)) = 2
POL(U81(x₁, x₂, x₃)) = 3 + (4)x_2 + (4)x_3
POL(n__zeros) = 0
POL(n__cons(x₁, x₂)) = (4)x_1 + (4)x_2
POL(isNatKind(x₁)) = 4
POL(U11(x₁, x₂)) = 2
POL(0) = 0
POL(U94(x₁, x₂)) = 0
POL(cons(x₁, x₂)) = (4)x_1 + (4)x_2
POL(U91(x₁, x₂, x₃)) = 0
POL(n__0) = 0
POL(U13(x₁)) = 0
POL(U86(x₁)) = 1
POL(U92(x₁, x₂, x₃)) = 0
POL(U84(x₁, x₂, x₃)) = 3 + (3)x_2
POL(length(x₁)) = 2

The value of delta used in the strict ordering is 6.
The following usable rules [17] were oriented:

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

                      ↳ Narrowing

                        ↳ 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

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ 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__cons(x0, x1)) → ISNATILISTKIND(n__cons(x0, x1))
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))

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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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__cons(x0, x1)) → ISNATILISTKIND(n__cons(x0, x1))
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))

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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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:

Semiunifier: [ ]
Matcher: [ ]

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

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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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

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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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

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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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__s(x0), y1) → U45¹(isNat(s(x0)), activate(y1))
U44¹(tt, n__0, y1) → U45¹(isNat(0), activate(y1))
U44¹(tt, n__cons(x0, x1), y1) → U45¹(isNat(cons(x0, x1)), activate(y1))
U44¹(tt, n__length(x0), y1) → U45¹(isNat(length(x0)), activate(y1))
U44¹(tt, n__zeros, y1) → U45¹(isNat(zeros), activate(y1))
U44¹(tt, n__nil, y1) → U45¹(isNat(nil), 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__s(x0), y1) → U45¹(isNat(s(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(x0, x1)), activate(y1))
U44¹(tt, n__length(x0), y1) → U45¹(isNat(length(x0)), activate(y1))
U44¹(tt, n__zeros, y1) → U45¹(isNat(zeros), activate(y1))
U44¹(tt, n__nil, y1) → U45¹(isNat(nil), 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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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__0) → ISNATILIST(0)
U45¹(tt, n__length(x0)) → ISNATILIST(length(x0))
U45¹(tt, n__s(x0)) → ISNATILIST(s(x0))
U45¹(tt, n__zeros) → ISNATILIST(zeros)
U45¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(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))
U44¹(tt, x0, y1) → U45¹(isNat(x0), activate(y1))
U42¹(tt, V1, V2) → U43¹(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U45¹(tt, n__s(x0)) → ISNATILIST(s(x0))
U45¹(tt, n__length(x0)) → ISNATILIST(length(x0))
U44¹(tt, n__s(x0), y1) → U45¹(isNat(s(x0)), activate(y1))
U45¹(tt, n__zeros) → ISNATILIST(zeros)
U44¹(tt, n__0, y1) → U45¹(isNat(0), activate(y1))
U45¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(x0, x1))
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(x0, x1)), activate(y1))
ISNATILIST(n__cons(V1, V2)) → U41¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U44¹(tt, n__length(x0), y1) → U45¹(isNat(length(x0)), activate(y1))
U44¹(tt, n__zeros, y1) → U45¹(isNat(zeros), activate(y1))
U45¹(tt, n__nil) → ISNATILIST(nil)
U44¹(tt, n__nil, y1) → U45¹(isNat(nil), activate(y1))
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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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__s(x0), y1) → U45¹(isNat(s(x0)), activate(y1)) at position [0] we obtained the following new rules:

U44¹(tt, n__s(x0), y1) → U45¹(isNat(n__s(x0)), activate(y1))

↳ 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(x0))
U45¹(tt, n__s(x0)) → ISNATILIST(s(x0))
U45¹(tt, n__zeros) → ISNATILIST(zeros)
U44¹(tt, n__0, y1) → U45¹(isNat(0), activate(y1))
U45¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(x0, x1))
U44¹(tt, n__s(x0), y1) → U45¹(isNat(n__s(x0)), activate(y1))
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(x0, x1)), activate(y1))
U44¹(tt, n__length(x0), y1) → U45¹(isNat(length(x0)), activate(y1))
U44¹(tt, n__zeros, y1) → U45¹(isNat(zeros), activate(y1))
U45¹(tt, n__nil) → ISNATILIST(nil)
U44¹(tt, n__nil, y1) → U45¹(isNat(nil), activate(y1))
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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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

                          ↳ 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__s(x0)) → ISNATILIST(s(x0))
U45¹(tt, n__length(x0)) → ISNATILIST(length(x0))
U45¹(tt, n__zeros) → ISNATILIST(zeros)
U45¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(x0, x1))
U41¹(tt, V1, V2) → U42¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U44¹(tt, n__s(x0), y1) → U45¹(isNat(n__s(x0)), activate(y1))
U45¹(tt, n__0) → ISNATILIST(0)
U44¹(tt, n__cons(x0, x1), y1) → U45¹(isNat(cons(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))
U44¹(tt, n__length(x0), y1) → U45¹(isNat(length(x0)), activate(y1))
U44¹(tt, n__zeros, y1) → U45¹(isNat(zeros), activate(y1))
U45¹(tt, n__nil) → ISNATILIST(nil)
U44¹(tt, n__nil, y1) → U45¹(isNat(nil), activate(y1))
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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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__cons(x0, x1), y1) → U45¹(isNat(cons(x0, x1)), activate(y1)) at position [0] we obtained the following new rules:

U44¹(tt, n__cons(x0, x1), y2) → U45¹(isNat(n__cons(x0, x1)), activate(y2))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ Narrowing

              ↳ 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))
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(x0))
U45¹(tt, n__s(x0)) → ISNATILIST(s(x0))
U45¹(tt, n__zeros) → ISNATILIST(zeros)
U44¹(tt, n__cons(x0, x1), y2) → U45¹(isNat(n__cons(x0, x1)), activate(y2))
U45¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(x0, x1))
U44¹(tt, n__s(x0), y1) → U45¹(isNat(n__s(x0)), activate(y1))
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__length(x0), y1) → U45¹(isNat(length(x0)), 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__nil) → ISNATILIST(nil)
U44¹(tt, n__nil, y1) → U45¹(isNat(nil), activate(y1))
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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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

                            ↳ 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(x0))
U45¹(tt, n__s(x0)) → ISNATILIST(s(x0))
U45¹(tt, n__zeros) → ISNATILIST(zeros)
U45¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(x0, x1))
U44¹(tt, n__s(x0), y1) → U45¹(isNat(n__s(x0)), activate(y1))
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__0, y0) → U45¹(isNat(n__0), activate(y0))
U44¹(tt, n__length(x0), y1) → U45¹(isNat(length(x0)), activate(y1))
U44¹(tt, n__zeros, y1) → U45¹(isNat(zeros), activate(y1))
U45¹(tt, n__nil) → ISNATILIST(nil)
U44¹(tt, n__nil, y1) → U45¹(isNat(nil), activate(y1))
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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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

                            ↳ 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))
U44¹(tt, x0, y1) → U45¹(isNat(x0), activate(y1))
U42¹(tt, V1, V2) → U43¹(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U45¹(tt, n__s(x0)) → ISNATILIST(s(x0))
U45¹(tt, n__length(x0)) → ISNATILIST(length(x0))
U45¹(tt, n__zeros) → ISNATILIST(zeros)
U45¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(x0, x1))
U45¹(tt, n__0) → ISNATILIST(n__0)
U41¹(tt, V1, V2) → U42¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U44¹(tt, n__s(x0), y1) → U45¹(isNat(n__s(x0)), activate(y1))
ISNATILIST(n__cons(V1, V2)) → U41¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U44¹(tt, n__length(x0), y1) → U45¹(isNat(length(x0)), 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__nil) → ISNATILIST(nil)
U44¹(tt, n__nil, y1) → U45¹(isNat(nil), activate(y1))
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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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

                            ↳ 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(x0))
U45¹(tt, n__s(x0)) → ISNATILIST(s(x0))
U45¹(tt, n__zeros) → ISNATILIST(zeros)
U45¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(x0, x1))
U44¹(tt, n__s(x0), y1) → U45¹(isNat(n__s(x0)), activate(y1))
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))
U44¹(tt, n__length(x0), y1) → U45¹(isNat(length(x0)), activate(y1))
U44¹(tt, n__zeros, y1) → U45¹(isNat(zeros), activate(y1))
U45¹(tt, n__nil) → ISNATILIST(nil)
U44¹(tt, n__nil, y1) → U45¹(isNat(nil), activate(y1))
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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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__s(x0)) → ISNATILIST(s(x0)) at position [0] we obtained the following new rules:

U45¹(tt, n__s(x0)) → ISNATILIST(n__s(x0))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ Narrowing

              ↳ 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))
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(x0))
U45¹(tt, n__zeros) → ISNATILIST(zeros)
U45¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(x0, x1))
U41¹(tt, V1, V2) → U42¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U44¹(tt, n__s(x0), y1) → U45¹(isNat(n__s(x0)), activate(y1))
ISNATILIST(n__cons(V1, V2)) → U41¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U44¹(tt, n__length(x0), y1) → U45¹(isNat(length(x0)), 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__s(x0)) → ISNATILIST(n__s(x0))
U45¹(tt, n__nil) → ISNATILIST(nil)
U44¹(tt, n__nil, y1) → U45¹(isNat(nil), activate(y1))
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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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

                            ↳ 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))
U42¹(tt, V1, V2) → U43¹(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U45¹(tt, n__length(x0)) → ISNATILIST(length(x0))
U45¹(tt, n__zeros) → ISNATILIST(zeros)
U45¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(x0, x1))
U44¹(tt, n__s(x0), y1) → U45¹(isNat(n__s(x0)), activate(y1))
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))
U44¹(tt, n__length(x0), y1) → U45¹(isNat(length(x0)), activate(y1))
U44¹(tt, n__zeros, y1) → U45¹(isNat(zeros), activate(y1))
U45¹(tt, n__nil) → ISNATILIST(nil)
U44¹(tt, n__nil, y1) → U45¹(isNat(nil), activate(y1))
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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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

                            ↳ 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))
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(x0))
U45¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(x0, x1))
U41¹(tt, V1, V2) → U42¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U44¹(tt, n__s(x0), y1) → U45¹(isNat(n__s(x0)), activate(y1))
ISNATILIST(n__cons(V1, V2)) → U41¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U44¹(tt, n__length(x0), y1) → U45¹(isNat(length(x0)), activate(y1))
U44¹(tt, n__0, y0) → U45¹(isNat(n__0), activate(y0))
U45¹(tt, n__zeros) → ISNATILIST(n__zeros)
U44¹(tt, n__zeros, y1) → U45¹(isNat(zeros), activate(y1))
U45¹(tt, n__nil) → ISNATILIST(nil)
U44¹(tt, n__nil, y1) → U45¹(isNat(nil), activate(y1))
U45¹(tt, x0) → ISNATILIST(x0)
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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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

                            ↳ 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))
U42¹(tt, V1, V2) → U43¹(isNatIListKind(activate(V2)), activate(V1), activate(V2))
U45¹(tt, n__length(x0)) → ISNATILIST(length(x0))
U45¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(x0, x1))
U44¹(tt, n__s(x0), y1) → U45¹(isNat(n__s(x0)), activate(y1))
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))
U44¹(tt, n__length(x0), y1) → U45¹(isNat(length(x0)), activate(y1))
U44¹(tt, n__zeros, y1) → U45¹(isNat(zeros), activate(y1))
U45¹(tt, n__nil) → ISNATILIST(nil)
U44¹(tt, n__nil, y1) → U45¹(isNat(nil), activate(y1))
U45¹(tt, x0) → ISNATILIST(x0)
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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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__cons(x0, x1)) → ISNATILIST(cons(x0, x1)) at position [0] we obtained the following new rules:

U45¹(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(x0, x1))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ Narrowing

              ↳ 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))
U44¹(tt, x0, y1) → U45¹(isNat(x0), activate(y1))
U45¹(tt, n__length(x0)) → ISNATILIST(length(x0))
U41¹(tt, V1, V2) → U42¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U44¹(tt, n__s(x0), y1) → U45¹(isNat(n__s(x0)), activate(y1))
ISNATILIST(n__cons(V1, V2)) → U41¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U44¹(tt, n__length(x0), y1) → U45¹(isNat(length(x0)), 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__nil) → ISNATILIST(nil)
U45¹(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(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))

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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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

                            ↳ 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))
U45¹(tt, n__nil) → ISNATILIST(n__nil)
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(x0))
U44¹(tt, n__s(x0), y1) → U45¹(isNat(n__s(x0)), activate(y1))
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))
U44¹(tt, n__length(x0), y1) → U45¹(isNat(length(x0)), activate(y1))
U44¹(tt, n__zeros, y1) → U45¹(isNat(zeros), activate(y1))
U44¹(tt, n__nil, y1) → U45¹(isNat(nil), activate(y1))
U45¹(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(x0, x1))
U45¹(tt, x0) → ISNATILIST(x0)
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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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

                            ↳ 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))
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(x0))
U41¹(tt, V1, V2) → U42¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U44¹(tt, n__s(x0), y1) → U45¹(isNat(n__s(x0)), 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))
U44¹(tt, n__length(x0), y1) → U45¹(isNat(length(x0)), activate(y1))
U44¹(tt, n__zeros, y1) → U45¹(isNat(zeros), activate(y1))
U45¹(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(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))

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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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

                            ↳ 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__zeros) → ISNATILIST(n__cons(0, n__zeros))
U45¹(tt, n__length(x0)) → ISNATILIST(length(x0))
U45¹(tt, n__zeros) → ISNATILIST(cons(n__0, n__zeros))
U44¹(tt, n__s(x0), y1) → U45¹(isNat(n__s(x0)), activate(y1))
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__length(x0), y1) → U45¹(isNat(length(x0)), activate(y1))
U44¹(tt, n__0, y0) → U45¹(isNat(n__0), activate(y0))
U44¹(tt, n__zeros, y1) → U45¹(isNat(zeros), activate(y1))
U44¹(tt, n__nil, y1) → U45¹(isNat(nil), activate(y1))
U45¹(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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

                            ↳ 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:

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__zeros) → ISNATILIST(n__cons(0, n__zeros))
U45¹(tt, n__length(x0)) → ISNATILIST(length(x0))
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__zeros) → ISNATILIST(cons(n__0, n__zeros))
U41¹(tt, V1, V2) → U42¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U44¹(tt, n__s(x0), y1) → U45¹(isNat(n__s(x0)), 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))
U44¹(tt, n__length(x0), y1) → U45¹(isNat(length(x0)), activate(y1))
U45¹(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(x0, x1))
U44¹(tt, n__nil, y1) → U45¹(isNat(nil), activate(y1))
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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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

                            ↳ 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__zeros) → ISNATILIST(n__cons(0, n__zeros))
U45¹(tt, n__length(x0)) → ISNATILIST(length(x0))
U44¹(tt, n__zeros, y0) → U45¹(isNat(cons(0, n__zeros)), activate(y0))
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__s(x0), y1) → U45¹(isNat(n__s(x0)), 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))
U44¹(tt, n__length(x0), y1) → U45¹(isNat(length(x0)), activate(y1))
U45¹(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(x0, x1))
U44¹(tt, n__nil, y1) → U45¹(isNat(nil), activate(y1))
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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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

                            ↳ 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

                                                                                        ↳ 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__zeros) → ISNATILIST(n__cons(0, n__zeros))
U45¹(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))
U45¹(tt, n__length(x0)) → ISNATILIST(length(x0))
U44¹(tt, n__zeros, y0) → U45¹(isNat(cons(0, n__zeros)), activate(y0))
U44¹(tt, n__s(x0), y1) → U45¹(isNat(n__s(x0)), activate(y1))
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__length(x0), y1) → U45¹(isNat(length(x0)), activate(y1))
U44¹(tt, n__0, y0) → U45¹(isNat(n__0), activate(y0))
U44¹(tt, n__nil, y1) → U45¹(isNat(nil), activate(y1))
U45¹(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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

                            ↳ 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

                                                                                        ↳ 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))
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))
U45¹(tt, n__length(x0)) → ISNATILIST(length(x0))
U45¹(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))
U44¹(tt, n__zeros, y0) → U45¹(isNat(cons(0, n__zeros)), activate(y0))
U44¹(tt, n__nil, y0) → U45¹(isNat(n__nil), activate(y0))
U41¹(tt, V1, V2) → U42¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U44¹(tt, n__s(x0), y1) → U45¹(isNat(n__s(x0)), 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))
U44¹(tt, n__length(x0), y1) → U45¹(isNat(length(x0)), activate(y1))
U45¹(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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

                            ↳ 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

                                                                                        ↳ 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__zeros) → ISNATILIST(n__cons(0, n__zeros))
U45¹(tt, n__length(x0)) → ISNATILIST(length(x0))
U45¹(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))
U44¹(tt, n__zeros, y0) → U45¹(isNat(cons(0, n__zeros)), activate(y0))
U41¹(tt, V1, V2) → U42¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U44¹(tt, n__s(x0), y1) → U45¹(isNat(n__s(x0)), activate(y1))
ISNATILIST(n__cons(V1, V2)) → U41¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U44¹(tt, n__length(x0), y1) → U45¹(isNat(length(x0)), activate(y1))
U44¹(tt, n__0, y0) → U45¹(isNat(n__0), activate(y0))
U45¹(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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

                            ↳ 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

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ 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))
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))
U45¹(tt, n__length(x0)) → ISNATILIST(length(x0))
U44¹(tt, n__s(x0), y1) → U45¹(isNat(n__s(x0)), activate(y1))
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__0, y0) → U45¹(isNat(n__0), activate(y0))
U44¹(tt, n__length(x0), y1) → U45¹(isNat(length(x0)), activate(y1))
U45¹(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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

                            ↳ 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

                                                                                        ↳ 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__zeros) → ISNATILIST(n__cons(0, n__zeros))
U45¹(tt, n__length(x0)) → ISNATILIST(length(x0))
U45¹(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))
U41¹(tt, V1, V2) → U42¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U44¹(tt, n__s(x0), y1) → U45¹(isNat(n__s(x0)), activate(y1))
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__length(x0), y1) → U45¹(isNat(length(x0)), activate(y1))
U44¹(tt, n__0, y0) → U45¹(isNat(n__0), activate(y0))
U45¹(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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

                            ↳ 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

                                                                                        ↳ 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))
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))
U45¹(tt, n__length(x0)) → ISNATILIST(length(x0))
U44¹(tt, n__s(x0), y1) → U45¹(isNat(n__s(x0)), activate(y1))
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))
ISNATILIST(n__cons(V1, V2)) → U41¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U44¹(tt, n__0, y0) → U45¹(isNat(n__0), activate(y0))
U44¹(tt, n__length(x0), y1) → U45¹(isNat(length(x0)), activate(y1))
U45¹(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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

                            ↳ 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

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ 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))
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__length(x0)) → ISNATILIST(length(x0))
U45¹(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))
U41¹(tt, V1, V2) → U42¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U44¹(tt, n__s(x0), y1) → U45¹(isNat(n__s(x0)), activate(y1))
ISNATILIST(n__cons(V1, V2)) → U41¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U44¹(tt, n__length(x0), y1) → U45¹(isNat(length(x0)), activate(y1))
U44¹(tt, n__0, y0) → U45¹(isNat(n__0), activate(y0))
U45¹(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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(x0)), activate(y1))

The remaining pairs can at least be oriented weakly.

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__zeros) → ISNATILIST(n__cons(0, n__zeros))
U45¹(tt, n__length(x0)) → ISNATILIST(length(x0))
U45¹(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))
U41¹(tt, V1, V2) → U42¹(isNatKind(activate(V1)), activate(V1), activate(V2))
U44¹(tt, n__s(x0), y1) → U45¹(isNat(n__s(x0)), 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(n__cons(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)) = 1 + 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)) = 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)) = 1 + 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)) = x₁
POL(n__zeros) = 0
POL(nil) = 0
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

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

                            ↳ 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

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ 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))
U44¹(tt, n__s(x0), y1) → U45¹(isNat(n__s(x0)), activate(y1))
U41¹(tt, V1, V2) → U42¹(isNatKind(activate(V1)), 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__zeros) → ISNATILIST(n__cons(0, n__zeros))
U45¹(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))
U45¹(tt, n__length(x0)) → ISNATILIST(length(x0))
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(n__cons(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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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(n__s(x0)), activate(y1))

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))
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))
U45¹(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))
U45¹(tt, n__length(x0)) → ISNATILIST(length(x0))
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(n__cons(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)) = 1 + x₂ + x₃
POL(U82(x₁, x₂, x₃)) = 1 + 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)) = 1 + 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:

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

                            ↳ 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

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ 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))
U44¹(tt, x0, y1) → U45¹(isNat(x0), activate(y1))
U42¹(tt, V1, V2) → U43¹(isNatIListKind(activate(V2)), activate(V1), activate(V2))
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__length(x0)) → ISNATILIST(length(x0))
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(n__cons(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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(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(x0))

The remaining pairs can at least be oriented weakly.

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))
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(n__cons(x0, x1))
U45¹(tt, x0) → ISNATILIST(x0)

Used ordering: Polynomial interpretation [25,35]:

POL(U43¹(x₁, x₂, x₃)) = (4)x_3
POL(U42¹(x₁, x₂, x₃)) = (4)x_3
POL(U82(x₁, x₂, x₃)) = (2)x_3
POL(ISNATILIST(x₁)) = (2)x_1
POL(U12(x₁, x₂)) = 0
POL(activate(x₁)) = x_1
POL(n__s(x₁)) = 0
POL(n__nil) = 0
POL(U51(x₁, x₂)) = 0
POL(U44¹(x₁, x₂, x₃)) = (4)x_3
POL(isNatIListKind(x₁)) = 0
POL(U21(x₁, x₂)) = 0
POL(U23(x₁)) = 0
POL(U45¹(x₁, x₂)) = (4)x_2
POL(tt) = 0
POL(U93(x₁, x₂, x₃)) = 3
POL(isNatList(x₁)) = (2)x_1
POL(zeros) = 0
POL(U52(x₁)) = 0
POL(U85(x₁, x₂)) = (2)x_2
POL(s(x₁)) = 0
POL(isNat(x₁)) = 0
POL(U71(x₁)) = 0
POL(nil) = 0
POL(U61(x₁)) = 0
POL(U83(x₁, x₂, x₃)) = (2)x_3
POL(n__length(x₁)) = 4 + x_1
POL(U22(x₁, x₂)) = 0
POL(U81(x₁, x₂, x₃)) = (4)x_3
POL(n__zeros) = 0
POL(n__cons(x₁, x₂)) = (4)x_2
POL(U11(x₁, x₂)) = 0
POL(isNatKind(x₁)) = 0
POL(0) = 0
POL(U41¹(x₁, x₂, x₃)) = (4)x_3
POL(U94(x₁, x₂)) = 3
POL(cons(x₁, x₂)) = (4)x_2
POL(U91(x₁, x₂, x₃)) = 4 + (3)x_2
POL(n__0) = 0
POL(U13(x₁)) = 0
POL(U86(x₁)) = 0
POL(U92(x₁, x₂, x₃)) = 3
POL(U84(x₁, x₂, x₃)) = (2)x_3
POL(length(x₁)) = 4 + x_1

The value of delta used in the strict ordering is 8.
The following usable rules [17] were oriented:

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

                            ↳ 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

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Narrowing

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ 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))
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))
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(n__cons(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(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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