Termination w.r.t. Q proof of /tmp/tmpq0Mkdn/LengthOfFiniteLists_complete

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

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

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

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

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

Q is empty.

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

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

The TRS R consists of the following rules:

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

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

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

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs with 27 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

          ↳ QDP

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

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

The TRS R consists of the following rules:

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

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

The following pairs can be oriented strictly and are deleted.

ISNAT(n__length(V1)) → ISNATILISTKIND(activate(V1))
ISNATKIND(n__length(V1)) → ACTIVATE(V1)
U61¹(tt, L) → ACTIVATE(L)
LENGTH(cons(N, L)) → AND(isNatList(activate(L)), n__isNatIListKind(activate(L)))
ISNAT(n__length(V1)) → U11¹(isNatIListKind(activate(V1)), activate(V1))
ISNATKIND(n__length(V1)) → ISNATILISTKIND(activate(V1))
LENGTH(cons(N, L)) → ACTIVATE(L)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
LENGTH(cons(N, L)) → AND(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N)))
ISNAT(n__length(V1)) → ACTIVATE(V1)
ACTIVATE(n__length(X)) → ACTIVATE(X)

The remaining pairs can at least be oriented weakly.

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

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

M( U52(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( n__isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( activate(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

/	1	0	\
\	0	1	/

x₂

M( n__nil ) =

/	0	\
\	1	/

M( n__s(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( U21(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	1	/

x₂

M( isNatIListKind(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( tt ) =

/	0	\
\	0	/

M( isNatList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( zeros ) =

/	0	\
\	0	/

M( U12(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( s(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( U51(x₁, ..., x₃) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

/	0	0	\
\	0	0	/

x₃

M( nil ) =

/	0	\
\	1	/

M( n__length(x₁) ) =

/	0	\
\	1	/

/	0	0	\
\	1	1	/

x₁

M( n__isNatKind(x₁) ) =

/	0	\
\	0	/

/	0	1	\
\	0	1	/

x₁

M( n__zeros ) =

/	0	\
\	0	/

M( n__cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	1	\
\	0	1	/

x₁

/	1	0	\
\	0	1	/

x₂

M( U11(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( isNatKind(x₁) ) =

/	0	\
\	0	/

/	0	1	\
\	0	1	/

x₁

M( 0 ) =

/	0	\
\	0	/

M( n__and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

/	1	0	\
\	0	1	/

x₂

M( cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	1	\
\	0	1	/

x₁

/	1	0	\
\	0	1	/

x₂

M( U22(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U53(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U61(x₁, x₂) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	1	1	/

x₂

M( n__0 ) =

/	0	\
\	0	/

M( n__isNatIListKind(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( length(x₁) ) =

/	0	\
\	1	/

/	0	0	\
\	1	1	/

x₁

Tuple symbols:

M( U51¹(x₁, ..., x₃) ) =

[

]

x₁

[

]

x₂

[

]

x₃

M( ISNATLIST(x₁) ) =

[

]

x₁

M( U52¹(x₁, x₂) ) =

[

]

x₁

[

]

x₂

M( LENGTH(x₁) ) =

[

]

x₁

M( ISNATILISTKIND(x₁) ) =

[

]

x₁

M( U11¹(x₁, x₂) ) =

[

]

x₁

[

]

x₂

M( ISNATKIND(x₁) ) =

[

]

x₁

M( AND(x₁, x₂) ) =

[

]

x₁

[

]

x₂

M( U61¹(x₁, x₂) ) =

[

]

x₁

[

]

x₂

M( U21¹(x₁, x₂) ) =

[

]

x₁

[

]

x₂

M( ACTIVATE(x₁) ) =

[

]

x₁

M( ISNAT(x₁) ) =

[

]

x₁

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

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

          ↳ QDP

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

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

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 3 SCCs with 11 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                      ↳ Narrowing

                    ↳ QDP

                    ↳ QDP

          ↳ QDP

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

U61¹(tt, L) → LENGTH(activate(L))
LENGTH(cons(N, L)) → U61¹(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U61¹(tt, L) → LENGTH(activate(L)) at position [0] we obtained the following new rules:

U61¹(tt, n__isNatKind(x0)) → LENGTH(isNatKind(x0))
U61¹(tt, n__length(x0)) → LENGTH(length(activate(x0)))
U61¹(tt, n__isNat(x0)) → LENGTH(isNat(x0))
U61¹(tt, n__and(x0, x1)) → LENGTH(and(activate(x0), x1))
U61¹(tt, n__0) → LENGTH(0)
U61¹(tt, n__isNatIListKind(x0)) → LENGTH(isNatIListKind(x0))
U61¹(tt, n__nil) → LENGTH(nil)
U61¹(tt, n__zeros) → LENGTH(zeros)
U61¹(tt, x0) → LENGTH(x0)
U61¹(tt, n__cons(x0, x1)) → LENGTH(cons(activate(x0), x1))
U61¹(tt, n__s(x0)) → LENGTH(s(activate(x0)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                    ↳ QDP

                    ↳ QDP

          ↳ QDP

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

U61¹(tt, n__length(x0)) → LENGTH(length(activate(x0)))
U61¹(tt, n__isNat(x0)) → LENGTH(isNat(x0))
U61¹(tt, n__and(x0, x1)) → LENGTH(and(activate(x0), x1))
U61¹(tt, n__0) → LENGTH(0)
LENGTH(cons(N, L)) → U61¹(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))
U61¹(tt, n__s(x0)) → LENGTH(s(activate(x0)))
U61¹(tt, n__isNatKind(x0)) → LENGTH(isNatKind(x0))
U61¹(tt, n__isNatIListKind(x0)) → LENGTH(isNatIListKind(x0))
U61¹(tt, n__nil) → LENGTH(nil)
U61¹(tt, n__zeros) → LENGTH(zeros)
U61¹(tt, x0) → LENGTH(x0)
U61¹(tt, n__cons(x0, x1)) → LENGTH(cons(activate(x0), x1))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U61¹(tt, n__0) → LENGTH(0) at position [0] we obtained the following new rules:

U61¹(tt, n__0) → LENGTH(n__0)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ DependencyGraphProof

                    ↳ QDP

                    ↳ QDP

          ↳ QDP

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

U61¹(tt, n__length(x0)) → LENGTH(length(activate(x0)))
U61¹(tt, n__0) → LENGTH(n__0)
U61¹(tt, n__isNat(x0)) → LENGTH(isNat(x0))
U61¹(tt, n__and(x0, x1)) → LENGTH(and(activate(x0), x1))
LENGTH(cons(N, L)) → U61¹(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))
U61¹(tt, n__s(x0)) → LENGTH(s(activate(x0)))
U61¹(tt, n__isNatKind(x0)) → LENGTH(isNatKind(x0))
U61¹(tt, n__nil) → LENGTH(nil)
U61¹(tt, n__isNatIListKind(x0)) → LENGTH(isNatIListKind(x0))
U61¹(tt, n__zeros) → LENGTH(zeros)
U61¹(tt, x0) → LENGTH(x0)
U61¹(tt, n__cons(x0, x1)) → LENGTH(cons(activate(x0), x1))

The TRS R consists of the following rules:

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

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

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                    ↳ QDP

                    ↳ QDP

          ↳ QDP

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

U61¹(tt, n__length(x0)) → LENGTH(length(activate(x0)))
U61¹(tt, n__isNatKind(x0)) → LENGTH(isNatKind(x0))
U61¹(tt, n__isNat(x0)) → LENGTH(isNat(x0))
U61¹(tt, n__nil) → LENGTH(nil)
U61¹(tt, n__isNatIListKind(x0)) → LENGTH(isNatIListKind(x0))
U61¹(tt, n__and(x0, x1)) → LENGTH(and(activate(x0), x1))
U61¹(tt, n__zeros) → LENGTH(zeros)
U61¹(tt, x0) → LENGTH(x0)
LENGTH(cons(N, L)) → U61¹(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))
U61¹(tt, n__cons(x0, x1)) → LENGTH(cons(activate(x0), x1))
U61¹(tt, n__s(x0)) → LENGTH(s(activate(x0)))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U61¹(tt, n__nil) → LENGTH(nil) at position [0] we obtained the following new rules:

U61¹(tt, n__nil) → LENGTH(n__nil)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                    ↳ QDP

                    ↳ QDP

          ↳ QDP

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

U61¹(tt, n__isNatKind(x0)) → LENGTH(isNatKind(x0))
U61¹(tt, n__length(x0)) → LENGTH(length(activate(x0)))
U61¹(tt, n__isNat(x0)) → LENGTH(isNat(x0))
U61¹(tt, n__and(x0, x1)) → LENGTH(and(activate(x0), x1))
U61¹(tt, n__isNatIListKind(x0)) → LENGTH(isNatIListKind(x0))
U61¹(tt, n__nil) → LENGTH(n__nil)
U61¹(tt, n__zeros) → LENGTH(zeros)
LENGTH(cons(N, L)) → U61¹(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))
U61¹(tt, x0) → LENGTH(x0)
U61¹(tt, n__cons(x0, x1)) → LENGTH(cons(activate(x0), x1))
U61¹(tt, n__s(x0)) → LENGTH(s(activate(x0)))

The TRS R consists of the following rules:

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

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

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                    ↳ QDP

                    ↳ QDP

          ↳ QDP

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

U61¹(tt, n__length(x0)) → LENGTH(length(activate(x0)))
U61¹(tt, n__isNatKind(x0)) → LENGTH(isNatKind(x0))
U61¹(tt, n__isNat(x0)) → LENGTH(isNat(x0))
U61¹(tt, n__isNatIListKind(x0)) → LENGTH(isNatIListKind(x0))
U61¹(tt, n__and(x0, x1)) → LENGTH(and(activate(x0), x1))
U61¹(tt, n__zeros) → LENGTH(zeros)
U61¹(tt, x0) → LENGTH(x0)
LENGTH(cons(N, L)) → U61¹(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))
U61¹(tt, n__cons(x0, x1)) → LENGTH(cons(activate(x0), x1))
U61¹(tt, n__s(x0)) → LENGTH(s(activate(x0)))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U61¹(tt, n__zeros) → LENGTH(zeros) at position [0] we obtained the following new rules:

U61¹(tt, n__zeros) → LENGTH(n__zeros)
U61¹(tt, n__zeros) → LENGTH(cons(0, n__zeros))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                    ↳ QDP

                    ↳ QDP

          ↳ QDP

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

U61¹(tt, n__isNatKind(x0)) → LENGTH(isNatKind(x0))
U61¹(tt, n__length(x0)) → LENGTH(length(activate(x0)))
U61¹(tt, n__isNat(x0)) → LENGTH(isNat(x0))
U61¹(tt, n__and(x0, x1)) → LENGTH(and(activate(x0), x1))
U61¹(tt, n__isNatIListKind(x0)) → LENGTH(isNatIListKind(x0))
LENGTH(cons(N, L)) → U61¹(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))
U61¹(tt, x0) → LENGTH(x0)
U61¹(tt, n__cons(x0, x1)) → LENGTH(cons(activate(x0), x1))
U61¹(tt, n__zeros) → LENGTH(cons(0, n__zeros))
U61¹(tt, n__zeros) → LENGTH(n__zeros)
U61¹(tt, n__s(x0)) → LENGTH(s(activate(x0)))

The TRS R consists of the following rules:

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

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

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ QDPOrderProof

                    ↳ QDP

                    ↳ QDP

          ↳ QDP

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

U61¹(tt, n__length(x0)) → LENGTH(length(activate(x0)))
U61¹(tt, n__isNatKind(x0)) → LENGTH(isNatKind(x0))
U61¹(tt, n__isNat(x0)) → LENGTH(isNat(x0))
U61¹(tt, n__isNatIListKind(x0)) → LENGTH(isNatIListKind(x0))
U61¹(tt, n__and(x0, x1)) → LENGTH(and(activate(x0), x1))
U61¹(tt, x0) → LENGTH(x0)
LENGTH(cons(N, L)) → U61¹(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))
U61¹(tt, n__cons(x0, x1)) → LENGTH(cons(activate(x0), x1))
U61¹(tt, n__zeros) → LENGTH(cons(0, n__zeros))
U61¹(tt, n__s(x0)) → LENGTH(s(activate(x0)))

The TRS R consists of the following rules:

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

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

The following pairs can be oriented strictly and are deleted.

U61¹(tt, n__length(x0)) → LENGTH(length(activate(x0)))

The remaining pairs can at least be oriented weakly.

U61¹(tt, n__isNatKind(x0)) → LENGTH(isNatKind(x0))
U61¹(tt, n__isNat(x0)) → LENGTH(isNat(x0))
U61¹(tt, n__isNatIListKind(x0)) → LENGTH(isNatIListKind(x0))
U61¹(tt, n__and(x0, x1)) → LENGTH(and(activate(x0), x1))
U61¹(tt, x0) → LENGTH(x0)
LENGTH(cons(N, L)) → U61¹(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))
U61¹(tt, n__cons(x0, x1)) → LENGTH(cons(activate(x0), x1))
U61¹(tt, n__zeros) → LENGTH(cons(0, n__zeros))
U61¹(tt, n__s(x0)) → LENGTH(s(activate(x0)))

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

M( U52(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( n__isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( activate(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	0	1	/

x₂

M( n__nil ) =

/	0	\
\	0	/

M( n__s(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U21(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( isNatIListKind(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( tt ) =

/	0	\
\	0	/

M( isNatList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( zeros ) =

/	0	\
\	0	/

M( U12(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( s(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U51(x₁, ..., x₃) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

/	0	0	\
\	0	0	/

x₃

M( nil ) =

/	0	\
\	0	/

M( n__length(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( n__isNatKind(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( n__zeros ) =

/	0	\
\	0	/

M( n__cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	1	1	/

x₂

M( U11(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( isNatKind(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( 0 ) =

/	0	\
\	0	/

M( n__and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	0	1	/

x₂

M( cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	1	1	/

x₂

M( U22(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U53(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U61(x₁, x₂) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( n__0 ) =

/	0	\
\	0	/

M( n__isNatIListKind(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( length(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

Tuple symbols:

M( LENGTH(x₁) ) =

[

]

x₁

M( U61¹(x₁, x₂) ) =

[

]

x₁

[

]

x₂

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ QDPOrderProof

                                                    ↳ QDP

                                                      ↳ QDPOrderProof

                    ↳ QDP

                    ↳ QDP

          ↳ QDP

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

U61¹(tt, n__isNatKind(x0)) → LENGTH(isNatKind(x0))
U61¹(tt, n__isNat(x0)) → LENGTH(isNat(x0))
U61¹(tt, n__and(x0, x1)) → LENGTH(and(activate(x0), x1))
U61¹(tt, n__isNatIListKind(x0)) → LENGTH(isNatIListKind(x0))
LENGTH(cons(N, L)) → U61¹(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))
U61¹(tt, x0) → LENGTH(x0)
U61¹(tt, n__cons(x0, x1)) → LENGTH(cons(activate(x0), x1))
U61¹(tt, n__zeros) → LENGTH(cons(0, n__zeros))
U61¹(tt, n__s(x0)) → LENGTH(s(activate(x0)))

The TRS R consists of the following rules:

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

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

The following pairs can be oriented strictly and are deleted.

U61¹(tt, n__s(x0)) → LENGTH(s(activate(x0)))

The remaining pairs can at least be oriented weakly.

U61¹(tt, n__isNatKind(x0)) → LENGTH(isNatKind(x0))
U61¹(tt, n__isNat(x0)) → LENGTH(isNat(x0))
U61¹(tt, n__and(x0, x1)) → LENGTH(and(activate(x0), x1))
U61¹(tt, n__isNatIListKind(x0)) → LENGTH(isNatIListKind(x0))
LENGTH(cons(N, L)) → U61¹(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))
U61¹(tt, x0) → LENGTH(x0)
U61¹(tt, n__cons(x0, x1)) → LENGTH(cons(activate(x0), x1))
U61¹(tt, n__zeros) → LENGTH(cons(0, n__zeros))

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

M( U52(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( n__isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( activate(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	0	1	/

x₂

M( n__nil ) =

/	0	\
\	0	/

M( n__s(x₁) ) =

/	1	\
\	1	/

/	0	0	\
\	0	0	/

x₁

M( U21(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( isNatIListKind(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( tt ) =

/	0	\
\	0	/

M( isNatList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( zeros ) =

/	0	\
\	0	/

M( U12(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( s(x₁) ) =

/	1	\
\	1	/

/	0	0	\
\	0	0	/

x₁

M( isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U51(x₁, ..., x₃) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

/	0	0	\
\	0	0	/

x₃

M( nil ) =

/	0	\
\	0	/

M( n__length(x₁) ) =

/	1	\
\	1	/

/	0	0	\
\	0	0	/

x₁

M( n__isNatKind(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( n__zeros ) =

/	0	\
\	0	/

M( n__cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	1	\
\	0	0	/

x₂

M( U11(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( isNatKind(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( 0 ) =

/	0	\
\	0	/

M( n__and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	0	1	/

x₂

M( cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	1	\
\	0	0	/

x₂

M( U22(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U53(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U61(x₁, x₂) ) =

/	1	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( n__0 ) =

/	0	\
\	0	/

M( n__isNatIListKind(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( length(x₁) ) =

/	1	\
\	1	/

/	0	0	\
\	0	0	/

x₁

Tuple symbols:

M( LENGTH(x₁) ) =

[

]

x₁

M( U61¹(x₁, x₂) ) =

[

]

x₁

[

]

x₂

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ QDPOrderProof

                                                    ↳ QDP

                                                      ↳ QDPOrderProof

                                                        ↳ QDP

                                                          ↳ QDPOrderProof

                    ↳ QDP

                    ↳ QDP

          ↳ QDP

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

U61¹(tt, n__isNatKind(x0)) → LENGTH(isNatKind(x0))
U61¹(tt, n__isNat(x0)) → LENGTH(isNat(x0))
U61¹(tt, n__isNatIListKind(x0)) → LENGTH(isNatIListKind(x0))
U61¹(tt, n__and(x0, x1)) → LENGTH(and(activate(x0), x1))
U61¹(tt, x0) → LENGTH(x0)
LENGTH(cons(N, L)) → U61¹(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))
U61¹(tt, n__cons(x0, x1)) → LENGTH(cons(activate(x0), x1))
U61¹(tt, n__zeros) → LENGTH(cons(0, n__zeros))

The TRS R consists of the following rules:

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

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

The following pairs can be oriented strictly and are deleted.

U61¹(tt, n__isNatKind(x0)) → LENGTH(isNatKind(x0))
U61¹(tt, n__isNatIListKind(x0)) → LENGTH(isNatIListKind(x0))

The remaining pairs can at least be oriented weakly.

U61¹(tt, n__isNat(x0)) → LENGTH(isNat(x0))
U61¹(tt, n__and(x0, x1)) → LENGTH(and(activate(x0), x1))
U61¹(tt, x0) → LENGTH(x0)
LENGTH(cons(N, L)) → U61¹(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))
U61¹(tt, n__cons(x0, x1)) → LENGTH(cons(activate(x0), x1))
U61¹(tt, n__zeros) → LENGTH(cons(0, n__zeros))

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

M( U52(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( n__isNat(x₁) ) =

/	0	\
\	0	/

/	0	1	\
\	1	0	/

x₁

M( activate(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	0	1	/

x₂

M( n__nil ) =

/	0	\
\	1	/

M( n__s(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

M( U21(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( isNatIListKind(x₁) ) =

/	1	\
\	1	/

/	0	0	\
\	0	0	/

x₁

M( tt ) =

/	0	\
\	0	/

M( isNatList(x₁) ) =

/	0	\
\	0	/

/	1	1	\
\	1	1	/

x₁

M( zeros ) =

/	0	\
\	0	/

M( U12(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( s(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

M( isNat(x₁) ) =

/	0	\
\	0	/

/	0	1	\
\	1	0	/

x₁

M( U51(x₁, ..., x₃) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

/	0	0	\
\	0	0	/

x₃

M( nil ) =

/	0	\
\	1	/

M( n__length(x₁) ) =

/	0	\
\	0	/

/	0	1	\
\	0	0	/

x₁

M( n__isNatKind(x₁) ) =

/	1	\
\	1	/

/	1	0	\
\	0	0	/

x₁

M( n__zeros ) =

/	0	\
\	0	/

M( n__cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	1	1	/

x₂

M( U11(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( isNatKind(x₁) ) =

/	1	\
\	1	/

/	1	0	\
\	0	0	/

x₁

M( 0 ) =

/	0	\
\	0	/

M( n__and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	0	1	/

x₂

M( cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	1	1	/

x₂

M( U22(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U53(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U61(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	1	\
\	0	0	/

x₂

M( n__0 ) =

/	0	\
\	0	/

M( n__isNatIListKind(x₁) ) =

/	1	\
\	1	/

/	0	0	\
\	0	0	/

x₁

M( length(x₁) ) =

/	0	\
\	0	/

/	0	1	\
\	0	0	/

x₁

Tuple symbols:

M( LENGTH(x₁) ) =

[

]

x₁

M( U61¹(x₁, x₂) ) =

[

]

x₁

[

]

x₂

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ QDPOrderProof

                                                    ↳ QDP

                                                      ↳ QDPOrderProof

                                                        ↳ QDP

                                                          ↳ QDPOrderProof

                                                            ↳ QDP

                                                              ↳ QDPOrderProof

                    ↳ QDP

                    ↳ QDP

          ↳ QDP

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

U61¹(tt, n__isNat(x0)) → LENGTH(isNat(x0))
U61¹(tt, n__and(x0, x1)) → LENGTH(and(activate(x0), x1))
LENGTH(cons(N, L)) → U61¹(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))
U61¹(tt, x0) → LENGTH(x0)
U61¹(tt, n__cons(x0, x1)) → LENGTH(cons(activate(x0), x1))
U61¹(tt, n__zeros) → LENGTH(cons(0, n__zeros))

The TRS R consists of the following rules:

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

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

The following pairs can be oriented strictly and are deleted.

U61¹(tt, n__isNat(x0)) → LENGTH(isNat(x0))

The remaining pairs can at least be oriented weakly.

U61¹(tt, n__and(x0, x1)) → LENGTH(and(activate(x0), x1))
LENGTH(cons(N, L)) → U61¹(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))
U61¹(tt, x0) → LENGTH(x0)
U61¹(tt, n__cons(x0, x1)) → LENGTH(cons(activate(x0), x1))
U61¹(tt, n__zeros) → LENGTH(cons(0, n__zeros))

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

M( U52(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( n__isNat(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( activate(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	1	\
\	0	0	/

x₁

/	1	0	\
\	0	1	/

x₂

M( n__nil ) =

/	0	\
\	0	/

M( n__s(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( U21(x₁, x₂) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( isNatIListKind(x₁) ) =

/	0	\
\	0	/

/	0	1	\
\	1	1	/

x₁

M( tt ) =

/	0	\
\	0	/

M( isNatList(x₁) ) =

/	0	\
\	0	/

/	0	1	\
\	0	0	/

x₁

M( zeros ) =

/	0	\
\	0	/

M( U12(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( s(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( isNat(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U51(x₁, ..., x₃) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

/	0	0	\
\	0	0	/

x₃

M( nil ) =

/	0	\
\	0	/

M( n__length(x₁) ) =

/	0	\
\	0	/

/	0	1	\
\	1	1	/

x₁

M( n__isNatKind(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( n__zeros ) =

/	0	\
\	0	/

M( n__cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	1	1	/

x₁

/	0	0	\
\	1	1	/

x₂

M( U11(x₁, x₂) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( isNatKind(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( 0 ) =

/	0	\
\	0	/

M( n__and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	1	\
\	0	0	/

x₁

/	1	0	\
\	0	1	/

x₂

M( cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	1	1	/

x₁

/	0	0	\
\	1	1	/

x₂

M( U22(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U53(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U61(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	1	\
\	1	1	/

x₂

M( n__0 ) =

/	0	\
\	0	/

M( n__isNatIListKind(x₁) ) =

/	0	\
\	0	/

/	0	1	\
\	1	1	/

x₁

M( length(x₁) ) =

/	0	\
\	0	/

/	0	1	\
\	1	1	/

x₁

Tuple symbols:

M( LENGTH(x₁) ) =

[

]

x₁

M( U61¹(x₁, x₂) ) =

[

]

x₁

[

]

x₂

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ QDPOrderProof

                                                    ↳ QDP

                                                      ↳ QDPOrderProof

                                                        ↳ QDP

                                                          ↳ QDPOrderProof

                                                            ↳ QDP

                                                              ↳ QDPOrderProof

                                                                ↳ QDP

                                                                  ↳ QDPOrderProof

                    ↳ QDP

                    ↳ QDP

          ↳ QDP

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

U61¹(tt, n__and(x0, x1)) → LENGTH(and(activate(x0), x1))
U61¹(tt, x0) → LENGTH(x0)
LENGTH(cons(N, L)) → U61¹(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))
U61¹(tt, n__cons(x0, x1)) → LENGTH(cons(activate(x0), x1))
U61¹(tt, n__zeros) → LENGTH(cons(0, n__zeros))

The TRS R consists of the following rules:

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

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

The following pairs can be oriented strictly and are deleted.

U61¹(tt, x0) → LENGTH(x0)

The remaining pairs can at least be oriented weakly.

U61¹(tt, n__and(x0, x1)) → LENGTH(and(activate(x0), x1))
LENGTH(cons(N, L)) → U61¹(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))
U61¹(tt, n__cons(x0, x1)) → LENGTH(cons(activate(x0), x1))
U61¹(tt, n__zeros) → LENGTH(cons(0, n__zeros))

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

M( U52(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	1	0	/

x₂

M( n__isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( activate(x₁) ) =

/	0	\
\	1	/

/	1	0	\
\	0	1	/

x₁

M( and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	1	1	/

x₁

/	1	0	\
\	0	1	/

x₂

M( n__nil ) =

/	1	\
\	0	/

M( n__s(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U21(x₁, x₂) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( isNatIListKind(x₁) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

M( tt ) =

/	0	\
\	1	/

M( isNatList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	1	0	/

x₁

M( zeros ) =

/	0	\
\	1	/

M( U12(x₁) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

M( s(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( isNat(x₁) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

M( U51(x₁, ..., x₃) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

/	0	0	\
\	1	0	/

x₃

M( nil ) =

/	1	\
\	0	/

M( n__length(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( n__isNatKind(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( n__zeros ) =

/	0	\
\	0	/

M( n__cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	1	1	/

x₂

M( U11(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

/	0	0	\
\	0	0	/

x₂

M( isNatKind(x₁) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

M( 0 ) =

/	0	\
\	0	/

M( n__and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	1	1	/

x₁

/	1	0	\
\	0	1	/

x₂

M( cons(x₁, x₂) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	1	1	/

x₂

M( U22(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( U53(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	1	1	/

x₁

M( U61(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( n__0 ) =

/	0	\
\	0	/

M( n__isNatIListKind(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( length(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

Tuple symbols:

M( LENGTH(x₁) ) =

[

]

x₁

M( U61¹(x₁, x₂) ) =

[

]

x₁

[

]

x₂

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ QDPOrderProof

                                                    ↳ QDP

                                                      ↳ QDPOrderProof

                                                        ↳ QDP

                                                          ↳ QDPOrderProof

                                                            ↳ QDP

                                                              ↳ QDPOrderProof

                                                                ↳ QDP

                                                                  ↳ QDPOrderProof

                                                                    ↳ QDP

                                                                      ↳ QDPOrderProof

                    ↳ QDP

                    ↳ QDP

          ↳ QDP

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

U61¹(tt, n__and(x0, x1)) → LENGTH(and(activate(x0), x1))
LENGTH(cons(N, L)) → U61¹(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))
U61¹(tt, n__cons(x0, x1)) → LENGTH(cons(activate(x0), x1))
U61¹(tt, n__zeros) → LENGTH(cons(0, n__zeros))

The TRS R consists of the following rules:

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

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

The following pairs can be oriented strictly and are deleted.

U61¹(tt, n__and(x0, x1)) → LENGTH(and(activate(x0), x1))

The remaining pairs can at least be oriented weakly.

LENGTH(cons(N, L)) → U61¹(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))
U61¹(tt, n__cons(x0, x1)) → LENGTH(cons(activate(x0), x1))
U61¹(tt, n__zeros) → LENGTH(cons(0, n__zeros))

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

M( U52(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( n__isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( activate(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	1	1	/

x₂

M( n__nil ) =

/	0	\
\	0	/

M( n__s(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U21(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( isNatIListKind(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( tt ) =

/	0	\
\	0	/

M( isNatList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( zeros ) =

/	1	\
\	0	/

M( U12(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( s(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U51(x₁, ..., x₃) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

/	0	0	\
\	0	0	/

x₃

M( nil ) =

/	0	\
\	0	/

M( n__length(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( n__isNatKind(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( n__zeros ) =

/	1	\
\	0	/

M( n__cons(x₁, x₂) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	1	\
\	0	1	/

x₂

M( U11(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( isNatKind(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( 0 ) =

/	0	\
\	0	/

M( n__and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	1	1	/

x₂

M( cons(x₁, x₂) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	1	\
\	0	1	/

x₂

M( U22(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U53(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U61(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( n__0 ) =

/	0	\
\	0	/

M( n__isNatIListKind(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( length(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

Tuple symbols:

M( LENGTH(x₁) ) =

[

]

x₁

M( U61¹(x₁, x₂) ) =

[

]

x₁

[

]

x₂

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ QDPOrderProof

                                                    ↳ QDP

                                                      ↳ QDPOrderProof

                                                        ↳ QDP

                                                          ↳ QDPOrderProof

                                                            ↳ QDP

                                                              ↳ QDPOrderProof

                                                                ↳ QDP

                                                                  ↳ QDPOrderProof

                                                                    ↳ QDP

                                                                      ↳ QDPOrderProof

                                                                        ↳ QDP

                                                                          ↳ QDPOrderProof

                    ↳ QDP

                    ↳ QDP

          ↳ QDP

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

LENGTH(cons(N, L)) → U61¹(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))
U61¹(tt, n__cons(x0, x1)) → LENGTH(cons(activate(x0), x1))
U61¹(tt, n__zeros) → LENGTH(cons(0, n__zeros))

The TRS R consists of the following rules:

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

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

The following pairs can be oriented strictly and are deleted.

U61¹(tt, n__cons(x0, x1)) → LENGTH(cons(activate(x0), x1))

The remaining pairs can at least be oriented weakly.

LENGTH(cons(N, L)) → U61¹(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))
U61¹(tt, n__zeros) → LENGTH(cons(0, n__zeros))

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

M( U52(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	1	0	/

x₂

M( n__isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( activate(x₁) ) =

/	0	\
\	1	/

/	1	0	\
\	0	1	/

x₁

M( and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

/	1	0	\
\	0	1	/

x₂

M( n__nil ) =

/	1	\
\	0	/

M( n__s(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U21(x₁, x₂) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( isNatIListKind(x₁) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

M( tt ) =

/	0	\
\	1	/

M( isNatList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	1	0	/

x₁

M( zeros ) =

/	0	\
\	1	/

M( U12(x₁) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

M( s(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( isNat(x₁) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

M( U51(x₁, ..., x₃) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

/	0	0	\
\	1	0	/

x₃

M( nil ) =

/	1	\
\	0	/

M( n__length(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( n__isNatKind(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( n__zeros ) =

/	0	\
\	0	/

M( n__cons(x₁, x₂) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	1	1	\
\	1	1	/

x₂

M( U11(x₁, x₂) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( isNatKind(x₁) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

M( 0 ) =

/	0	\
\	0	/

M( n__and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

/	1	0	\
\	0	1	/

x₂

M( cons(x₁, x₂) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	1	1	\
\	1	1	/

x₂

M( U22(x₁) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

M( U53(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( U61(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( n__0 ) =

/	0	\
\	0	/

M( n__isNatIListKind(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( length(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

Tuple symbols:

M( LENGTH(x₁) ) =

[

]

x₁

M( U61¹(x₁, x₂) ) =

[

]

x₁

[

]

x₂

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ QDPOrderProof

                                                    ↳ QDP

                                                      ↳ QDPOrderProof

                                                        ↳ QDP

                                                          ↳ QDPOrderProof

                                                            ↳ QDP

                                                              ↳ QDPOrderProof

                                                                ↳ QDP

                                                                  ↳ QDPOrderProof

                                                                    ↳ QDP

                                                                      ↳ QDPOrderProof

                                                                        ↳ QDP

                                                                          ↳ QDPOrderProof

                                                                            ↳ QDP

                                                                              ↳ QDPOrderProof

                    ↳ QDP

                    ↳ QDP

          ↳ QDP

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

LENGTH(cons(N, L)) → U61¹(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))
U61¹(tt, n__zeros) → LENGTH(cons(0, n__zeros))

The TRS R consists of the following rules:

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

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

The following pairs can be oriented strictly and are deleted.

U61¹(tt, n__zeros) → LENGTH(cons(0, n__zeros))

The remaining pairs can at least be oriented weakly.

LENGTH(cons(N, L)) → U61¹(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))

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

M( U52(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	1	\
\	0	0	/

x₂

M( n__isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( activate(x₁) ) =

/	1	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( and(x₁, x₂) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

/	1	0	\
\	0	1	/

x₂

M( n__nil ) =

/	0	\
\	1	/

M( n__s(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U21(x₁, x₂) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( isNatIListKind(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( tt ) =

/	1	\
\	0	/

M( isNatList(x₁) ) =

/	0	\
\	0	/

/	0	1	\
\	0	0	/

x₁

M( zeros ) =

/	1	\
\	0	/

M( U12(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( s(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( isNat(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U51(x₁, ..., x₃) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

/	0	1	\
\	0	0	/

x₃

M( nil ) =

/	0	\
\	1	/

M( n__length(x₁) ) =

/	1	\
\	1	/

/	0	0	\
\	0	0	/

x₁

M( n__isNatKind(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( n__zeros ) =

/	0	\
\	0	/

M( n__cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	1	/

x₂

M( U11(x₁, x₂) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( isNatKind(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( 0 ) =

/	0	\
\	0	/

M( n__and(x₁, x₂) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

/	1	0	\
\	0	1	/

x₂

M( cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	1	/

x₂

M( U22(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U53(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

M( U61(x₁, x₂) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( n__0 ) =

/	0	\
\	0	/

M( n__isNatIListKind(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( length(x₁) ) =

/	1	\
\	1	/

/	0	0	\
\	0	0	/

x₁

Tuple symbols:

M( LENGTH(x₁) ) =

[

]

x₁

M( U61¹(x₁, x₂) ) =

[

]

x₁

[

]

x₂

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ Narrowing

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ QDPOrderProof

                                                    ↳ QDP

                                                      ↳ QDPOrderProof

                                                        ↳ QDP

                                                          ↳ QDPOrderProof

                                                            ↳ QDP

                                                              ↳ QDPOrderProof

                                                                ↳ QDP

                                                                  ↳ QDPOrderProof

                                                                    ↳ QDP

                                                                      ↳ QDPOrderProof

                                                                        ↳ QDP

                                                                          ↳ QDPOrderProof

                                                                            ↳ QDP

                                                                              ↳ QDPOrderProof

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                    ↳ QDP

                    ↳ QDP

          ↳ QDP

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

LENGTH(cons(N, L)) → U61¹(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 1 less node.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ QDPOrderProof

                    ↳ QDP

          ↳ QDP

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

ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
ISNAT(n__s(V1)) → ACTIVATE(V1)
AND(tt, X) → ACTIVATE(X)
U21¹(tt, V1) → ACTIVATE(V1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ISNATILISTKIND(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ISNATILISTKIND(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ACTIVATE(n__isNat(X)) → ISNAT(X)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNAT(n__s(V1)) → U21¹(isNatKind(activate(V1)), activate(V1))
U21¹(tt, V1) → ISNAT(activate(V1))
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V1)

The TRS R consists of the following rules:

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

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

The following pairs can be oriented strictly and are deleted.

ACTIVATE(n__isNat(X)) → ISNAT(X)

The remaining pairs can at least be oriented weakly.

ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
ISNAT(n__s(V1)) → ACTIVATE(V1)
AND(tt, X) → ACTIVATE(X)
U21¹(tt, V1) → ACTIVATE(V1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ISNATILISTKIND(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ISNATILISTKIND(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNAT(n__s(V1)) → U21¹(isNatKind(activate(V1)), activate(V1))
U21¹(tt, V1) → ISNAT(activate(V1))
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V1)

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(ACTIVATE(x₁)) = x₁
POL(AND(x₁, x₂)) = x₂
POL(ISNAT(x₁)) = x₁
POL(ISNATILISTKIND(x₁)) = x₁
POL(ISNATKIND(x₁)) = x₁
POL(U11(x₁, x₂)) = x₁
POL(U12(x₁)) = 0
POL(U21(x₁, x₂)) = 0
POL(U21¹(x₁, x₂)) = x₂
POL(U22(x₁)) = 0
POL(U51(x₁, x₂, x₃)) = 0
POL(U52(x₁, x₂)) = 0
POL(U53(x₁)) = 0
POL(U61(x₁, x₂)) = x₂
POL(activate(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = 1 + x₁
POL(isNatIListKind(x₁)) = x₁
POL(isNatKind(x₁)) = x₁
POL(isNatList(x₁)) = 1 + x₁
POL(length(x₁)) = x₁
POL(n__0) = 0
POL(n__and(x₁, x₂)) = x₁ + x₂
POL(n__cons(x₁, x₂)) = x₁ + x₂
POL(n__isNat(x₁)) = 1 + x₁
POL(n__isNatIListKind(x₁)) = x₁
POL(n__isNatKind(x₁)) = x₁
POL(n__length(x₁)) = 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__isNat(X)) → isNat(X)
activate(X) → X
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U52(tt, V2) → U53(isNatList(activate(V2)))
U53(tt) → tt
U61(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
isNatKind(n__0) → tt
zeros → n__zeros
length(cons(N, L)) → U61(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))
length(nil) → 0
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNatIListKind(n__zeros) → tt
isNatIListKind(n__nil) → tt
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
and(tt, X) → activate(X)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__nil) → nil
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
0 → n__0
length(X) → n__length(X)
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                    ↳ QDP

          ↳ QDP

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

ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
ISNAT(n__s(V1)) → ACTIVATE(V1)
AND(tt, X) → ACTIVATE(X)
U21¹(tt, V1) → ACTIVATE(V1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ISNATILISTKIND(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ISNATILISTKIND(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
U21¹(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → U21¹(isNatKind(activate(V1)), activate(V1))
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V1)

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs with 3 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                                ↳ QDPOrderProof

                              ↳ QDP

                    ↳ QDP

          ↳ QDP

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

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

The TRS R consists of the following rules:

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

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

The following pairs can be oriented strictly and are deleted.

ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATILISTKIND(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V1)

The remaining pairs can at least be oriented weakly.

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

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(ACTIVATE(x₁)) = x₁
POL(AND(x₁, x₂)) = x₂
POL(ISNATILISTKIND(x₁)) = 1 + x₁
POL(ISNATKIND(x₁)) = x₁
POL(U11(x₁, x₂)) = 0
POL(U12(x₁)) = 0
POL(U21(x₁, x₂)) = 0
POL(U22(x₁)) = 0
POL(U51(x₁, x₂, x₃)) = x₃
POL(U52(x₁, x₂)) = 0
POL(U53(x₁)) = 0
POL(U61(x₁, x₂)) = 1 + x₂
POL(activate(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = x₁
POL(isNatIListKind(x₁)) = 1 + x₁
POL(isNatKind(x₁)) = x₁
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = 1 + x₁
POL(n__0) = 0
POL(n__and(x₁, x₂)) = x₁ + x₂
POL(n__cons(x₁, x₂)) = x₁ + x₂
POL(n__isNat(x₁)) = x₁
POL(n__isNatIListKind(x₁)) = 1 + x₁
POL(n__isNatKind(x₁)) = x₁
POL(n__length(x₁)) = 1 + x₁
POL(n__nil) = 1
POL(n__s(x₁)) = x₁
POL(n__zeros) = 0
POL(nil) = 1
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                              ↳ QDP

                    ↳ QDP

          ↳ QDP

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

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

The TRS R consists of the following rules:

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

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

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.

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

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

M( U52(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( n__isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( activate(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	1	1	/

x₁

/	1	0	\
\	0	1	/

x₂

M( n__nil ) =

/	1	\
\	0	/

M( n__s(x₁) ) =

/	0	\
\	0	/

/	1	1	\
\	0	0	/

x₁

M( U21(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( isNatIListKind(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	1	0	/

x₁

M( tt ) =

/	0	\
\	0	/

M( isNatList(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

M( zeros ) =

/	0	\
\	1	/

M( U12(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( s(x₁) ) =

/	0	\
\	0	/

/	1	1	\
\	0	0	/

x₁

M( isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U51(x₁, ..., x₃) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	0	0	/

x₂

/	0	0	\
\	0	0	/

x₃

M( nil ) =

/	1	\
\	0	/

M( n__length(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

M( n__isNatKind(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	1	0	/

x₁

M( n__zeros ) =

/	0	\
\	1	/

M( n__cons(x₁, x₂) ) =

/	0	\
\	1	/

/	1	0	\
\	0	1	/

x₁

/	1	0	\
\	0	0	/

x₂

M( U11(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( isNatKind(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	1	0	/

x₁

M( 0 ) =

/	0	\
\	0	/

M( n__and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	1	1	/

x₁

/	1	0	\
\	0	1	/

x₂

M( cons(x₁, x₂) ) =

/	0	\
\	1	/

/	1	0	\
\	0	1	/

x₁

/	1	0	\
\	0	0	/

x₂

M( U22(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U53(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U61(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	0	0	/

x₂

M( n__0 ) =

/	0	\
\	0	/

M( n__isNatIListKind(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	1	0	/

x₁

M( length(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

Tuple symbols:

M( ISNATILISTKIND(x₁) ) =

[

]

x₁

M( AND(x₁, x₂) ) =

[

]

x₁

[

]

x₂

M( ISNATKIND(x₁) ) =

[

]

x₁

M( ACTIVATE(x₁) ) =

[

]

x₁

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ Narrowing

                              ↳ QDP

                    ↳ QDP

          ↳ QDP

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

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

The TRS R consists of the following rules:

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

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

ISNATILISTKIND(n__cons(n__nil, y1)) → AND(isNatKind(nil), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__zeros, y1)) → AND(isNatKind(zeros), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(x0, y1)) → AND(isNatKind(x0), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__isNat(x0), y1)) → AND(isNatKind(isNat(x0)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → AND(isNatKind(length(activate(x0))), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__and(x0, x1), y1)) → AND(isNatKind(and(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → AND(isNatKind(cons(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(y0, y1)) → AND(n__isNatKind(activate(y0)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__isNatIListKind(x0), y1)) → AND(isNatKind(isNatIListKind(x0)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__isNatKind(x0), y1)) → AND(isNatKind(isNatKind(x0)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__0, y1)) → AND(isNatKind(0), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → AND(isNatKind(s(activate(x0))), n__isNatIListKind(activate(y1)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                              ↳ QDP

                    ↳ QDP

          ↳ QDP

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

ISNATILISTKIND(n__cons(n__nil, y1)) → AND(isNatKind(nil), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__zeros, y1)) → AND(isNatKind(zeros), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
ISNATILISTKIND(n__cons(n__and(x0, x1), y1)) → AND(isNatKind(and(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → AND(isNatKind(cons(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__isNatIListKind(x0), y1)) → AND(isNatKind(isNatIListKind(x0)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__isNatKind(x0), y1)) → AND(isNatKind(isNatKind(x0)), n__isNatIListKind(activate(y1)))
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ISNATILISTKIND(n__cons(x0, y1)) → AND(isNatKind(x0), n__isNatIListKind(activate(y1)))
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ISNATILISTKIND(n__cons(n__length(x0), y1)) → AND(isNatKind(length(activate(x0))), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__isNat(x0), y1)) → AND(isNatKind(isNat(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATILISTKIND(n__cons(y0, y1)) → AND(n__isNatKind(activate(y0)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → AND(isNatKind(s(activate(x0))), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__0, y1)) → AND(isNatKind(0), n__isNatIListKind(activate(y1)))

The TRS R consists of the following rules:

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

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

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                              ↳ QDP

                    ↳ QDP

          ↳ QDP

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

ISNATILISTKIND(n__cons(n__nil, y1)) → AND(isNatKind(nil), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__zeros, y1)) → AND(isNatKind(zeros), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2)
ISNATILISTKIND(n__cons(n__and(x0, x1), y1)) → AND(isNatKind(and(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → AND(isNatKind(cons(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__isNatIListKind(x0), y1)) → AND(isNatKind(isNatIListKind(x0)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__isNatKind(x0), y1)) → AND(isNatKind(isNatKind(x0)), n__isNatIListKind(activate(y1)))
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ISNATILISTKIND(n__cons(x0, y1)) → AND(isNatKind(x0), n__isNatIListKind(activate(y1)))
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ISNATILISTKIND(n__cons(n__isNat(x0), y1)) → AND(isNatKind(isNat(x0)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → AND(isNatKind(length(activate(x0))), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → AND(isNatKind(s(activate(x0))), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__0, y1)) → AND(isNatKind(0), n__isNatIListKind(activate(y1)))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ACTIVATE(n__and(X1, X2)) → AND(activate(X1), X2) at position [0] we obtained the following new rules:

ACTIVATE(n__and(n__isNatKind(x0), y1)) → AND(isNatKind(x0), y1)
ACTIVATE(n__and(n__cons(x0, x1), y1)) → AND(cons(activate(x0), x1), y1)
ACTIVATE(n__and(n__isNat(x0), y1)) → AND(isNat(x0), y1)
ACTIVATE(n__and(n__s(x0), y1)) → AND(s(activate(x0)), y1)
ACTIVATE(n__and(n__isNatIListKind(x0), y1)) → AND(isNatIListKind(x0), y1)
ACTIVATE(n__and(n__zeros, y1)) → AND(zeros, y1)
ACTIVATE(n__and(n__length(x0), y1)) → AND(length(activate(x0)), y1)
ACTIVATE(n__and(n__nil, y1)) → AND(nil, y1)
ACTIVATE(n__and(n__and(x0, x1), y1)) → AND(and(activate(x0), x1), y1)
ACTIVATE(n__and(x0, y1)) → AND(x0, y1)
ACTIVATE(n__and(n__0, y1)) → AND(0, y1)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Narrowing

                              ↳ QDP

                    ↳ QDP

          ↳ QDP

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

ACTIVATE(n__and(n__isNatKind(x0), y1)) → AND(isNatKind(x0), y1)
ISNATILISTKIND(n__cons(n__isNatKind(x0), y1)) → AND(isNatKind(isNatKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__zeros, y1)) → AND(zeros, y1)
ACTIVATE(n__and(n__length(x0), y1)) → AND(length(activate(x0)), y1)
ACTIVATE(n__and(n__and(x0, x1), y1)) → AND(and(activate(x0), x1), y1)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ISNATILISTKIND(n__cons(x0, y1)) → AND(isNatKind(x0), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → AND(isNatKind(length(activate(x0))), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__isNat(x0), y1)) → AND(isNat(x0), y1)
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ACTIVATE(n__and(n__nil, y1)) → AND(nil, y1)
ISNATILISTKIND(n__cons(n__s(x0), y1)) → AND(isNatKind(s(activate(x0))), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__0, y1)) → AND(0, y1)
ISNATILISTKIND(n__cons(n__nil, y1)) → AND(isNatKind(nil), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__zeros, y1)) → AND(isNatKind(zeros), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__cons(x0, x1), y1)) → AND(cons(activate(x0), x1), y1)
ISNATILISTKIND(n__cons(n__and(x0, x1), y1)) → AND(isNatKind(and(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → AND(isNatKind(cons(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__isNatIListKind(x0), y1)) → AND(isNatKind(isNatIListKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__s(x0), y1)) → AND(s(activate(x0)), y1)
ACTIVATE(n__and(n__isNatIListKind(x0), y1)) → AND(isNatIListKind(x0), y1)
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(x0, y1)) → AND(x0, y1)
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ISNATILISTKIND(n__cons(n__isNat(x0), y1)) → AND(isNatKind(isNat(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATILISTKIND(n__cons(n__0, y1)) → AND(isNatKind(0), n__isNatIListKind(activate(y1)))

The TRS R consists of the following rules:

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

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

ISNATKIND(n__s(n__isNat(x0))) → ISNATKIND(isNat(x0))
ISNATKIND(n__s(n__zeros)) → ISNATKIND(zeros)
ISNATKIND(n__s(x0)) → ISNATKIND(x0)
ISNATKIND(n__s(n__length(x0))) → ISNATKIND(length(activate(x0)))
ISNATKIND(n__s(n__s(x0))) → ISNATKIND(s(activate(x0)))
ISNATKIND(n__s(n__cons(x0, x1))) → ISNATKIND(cons(activate(x0), x1))
ISNATKIND(n__s(n__isNatKind(x0))) → ISNATKIND(isNatKind(x0))
ISNATKIND(n__s(n__nil)) → ISNATKIND(nil)
ISNATKIND(n__s(n__0)) → ISNATKIND(0)
ISNATKIND(n__s(n__and(x0, x1))) → ISNATKIND(and(activate(x0), x1))
ISNATKIND(n__s(n__isNatIListKind(x0))) → ISNATKIND(isNatIListKind(x0))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ Narrowing

                              ↳ QDP

                    ↳ QDP

          ↳ QDP

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

ISNATKIND(n__s(n__isNat(x0))) → ISNATKIND(isNat(x0))
ACTIVATE(n__and(n__isNatKind(x0), y1)) → AND(isNatKind(x0), y1)
ISNATKIND(n__s(x0)) → ISNATKIND(x0)
ISNATKIND(n__s(n__cons(x0, x1))) → ISNATKIND(cons(activate(x0), x1))
ACTIVATE(n__and(n__zeros, y1)) → AND(zeros, y1)
ISNATILISTKIND(n__cons(n__isNatKind(x0), y1)) → AND(isNatKind(isNatKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__length(x0), y1)) → AND(length(activate(x0)), y1)
ACTIVATE(n__and(n__and(x0, x1), y1)) → AND(and(activate(x0), x1), y1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ISNATILISTKIND(n__cons(x0, y1)) → AND(isNatKind(x0), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__isNat(x0), y1)) → AND(isNat(x0), y1)
ISNATILISTKIND(n__cons(n__length(x0), y1)) → AND(isNatKind(length(activate(x0))), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATKIND(n__s(n__isNatKind(x0))) → ISNATKIND(isNatKind(x0))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → AND(isNatKind(s(activate(x0))), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__nil, y1)) → AND(nil, y1)
ISNATKIND(n__s(n__and(x0, x1))) → ISNATKIND(and(activate(x0), x1))
ISNATKIND(n__s(n__isNatIListKind(x0))) → ISNATKIND(isNatIListKind(x0))
ACTIVATE(n__and(n__0, y1)) → AND(0, y1)
ISNATILISTKIND(n__cons(n__zeros, y1)) → AND(isNatKind(zeros), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__nil, y1)) → AND(isNatKind(nil), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__cons(x0, x1), y1)) → AND(cons(activate(x0), x1), y1)
ISNATKIND(n__s(n__s(x0))) → ISNATKIND(s(activate(x0)))
ISNATKIND(n__s(n__length(x0))) → ISNATKIND(length(activate(x0)))
ISNATILISTKIND(n__cons(n__and(x0, x1), y1)) → AND(isNatKind(and(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → AND(isNatKind(cons(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__isNatIListKind(x0), y1)) → AND(isNatKind(isNatIListKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__isNatIListKind(x0), y1)) → AND(isNatIListKind(x0), y1)
ACTIVATE(n__and(n__s(x0), y1)) → AND(s(activate(x0)), y1)
ISNATKIND(n__s(n__nil)) → ISNATKIND(nil)
AND(tt, X) → ACTIVATE(X)
ISNATKIND(n__s(n__0)) → ISNATKIND(0)
ACTIVATE(n__and(x0, y1)) → AND(x0, y1)
ISNATKIND(n__s(n__zeros)) → ISNATKIND(zeros)
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ISNATILISTKIND(n__cons(n__isNat(x0), y1)) → AND(isNatKind(isNat(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATILISTKIND(n__cons(n__0, y1)) → AND(isNatKind(0), n__isNatIListKind(activate(y1)))

The TRS R consists of the following rules:

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

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

ACTIVATE(n__and(n__zeros, y0)) → AND(n__zeros, y0)
ACTIVATE(n__and(n__zeros, y0)) → AND(cons(0, n__zeros), y0)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                              ↳ QDP

                    ↳ QDP

          ↳ QDP

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

ISNATKIND(n__s(n__isNat(x0))) → ISNATKIND(isNat(x0))
ACTIVATE(n__and(n__isNatKind(x0), y1)) → AND(isNatKind(x0), y1)
ISNATKIND(n__s(x0)) → ISNATKIND(x0)
ISNATKIND(n__s(n__cons(x0, x1))) → ISNATKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(n__isNatKind(x0), y1)) → AND(isNatKind(isNatKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__zeros, y0)) → AND(n__zeros, y0)
ACTIVATE(n__and(n__length(x0), y1)) → AND(length(activate(x0)), y1)
ACTIVATE(n__and(n__and(x0, x1), y1)) → AND(and(activate(x0), x1), y1)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ISNATILISTKIND(n__cons(x0, y1)) → AND(isNatKind(x0), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → AND(isNatKind(length(activate(x0))), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__isNat(x0), y1)) → AND(isNat(x0), y1)
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATKIND(n__s(n__isNatKind(x0))) → ISNATKIND(isNatKind(x0))
ACTIVATE(n__and(n__nil, y1)) → AND(nil, y1)
ISNATILISTKIND(n__cons(n__s(x0), y1)) → AND(isNatKind(s(activate(x0))), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__zeros, y0)) → AND(cons(0, n__zeros), y0)
ISNATKIND(n__s(n__and(x0, x1))) → ISNATKIND(and(activate(x0), x1))
ISNATKIND(n__s(n__isNatIListKind(x0))) → ISNATKIND(isNatIListKind(x0))
ACTIVATE(n__and(n__0, y1)) → AND(0, y1)
ISNATILISTKIND(n__cons(n__nil, y1)) → AND(isNatKind(nil), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__zeros, y1)) → AND(isNatKind(zeros), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__cons(x0, x1), y1)) → AND(cons(activate(x0), x1), y1)
ISNATKIND(n__s(n__length(x0))) → ISNATKIND(length(activate(x0)))
ISNATKIND(n__s(n__s(x0))) → ISNATKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__and(x0, x1), y1)) → AND(isNatKind(and(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → AND(isNatKind(cons(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__isNatIListKind(x0), y1)) → AND(isNatKind(isNatIListKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__s(x0), y1)) → AND(s(activate(x0)), y1)
ACTIVATE(n__and(n__isNatIListKind(x0), y1)) → AND(isNatIListKind(x0), y1)
AND(tt, X) → ACTIVATE(X)
ISNATKIND(n__s(n__nil)) → ISNATKIND(nil)
ISNATKIND(n__s(n__0)) → ISNATKIND(0)
ACTIVATE(n__and(x0, y1)) → AND(x0, y1)
ISNATKIND(n__s(n__zeros)) → ISNATKIND(zeros)
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ISNATILISTKIND(n__cons(n__isNat(x0), y1)) → AND(isNatKind(isNat(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATILISTKIND(n__cons(n__0, y1)) → AND(isNatKind(0), n__isNatIListKind(activate(y1)))

The TRS R consists of the following rules:

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

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

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                              ↳ QDP

                    ↳ QDP

          ↳ QDP

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

ISNATKIND(n__s(n__isNat(x0))) → ISNATKIND(isNat(x0))
ACTIVATE(n__and(n__isNatKind(x0), y1)) → AND(isNatKind(x0), y1)
ISNATKIND(n__s(x0)) → ISNATKIND(x0)
ISNATKIND(n__s(n__cons(x0, x1))) → ISNATKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(n__isNatKind(x0), y1)) → AND(isNatKind(isNatKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__length(x0), y1)) → AND(length(activate(x0)), y1)
ACTIVATE(n__and(n__and(x0, x1), y1)) → AND(and(activate(x0), x1), y1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ISNATILISTKIND(n__cons(x0, y1)) → AND(isNatKind(x0), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → AND(isNatKind(length(activate(x0))), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__isNat(x0), y1)) → AND(isNat(x0), y1)
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATKIND(n__s(n__isNatKind(x0))) → ISNATKIND(isNatKind(x0))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → AND(isNatKind(s(activate(x0))), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__nil, y1)) → AND(nil, y1)
ISNATKIND(n__s(n__and(x0, x1))) → ISNATKIND(and(activate(x0), x1))
ACTIVATE(n__and(n__zeros, y0)) → AND(cons(0, n__zeros), y0)
ISNATKIND(n__s(n__isNatIListKind(x0))) → ISNATKIND(isNatIListKind(x0))
ACTIVATE(n__and(n__0, y1)) → AND(0, y1)
ISNATILISTKIND(n__cons(n__zeros, y1)) → AND(isNatKind(zeros), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__nil, y1)) → AND(isNatKind(nil), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__cons(x0, x1), y1)) → AND(cons(activate(x0), x1), y1)
ISNATKIND(n__s(n__s(x0))) → ISNATKIND(s(activate(x0)))
ISNATKIND(n__s(n__length(x0))) → ISNATKIND(length(activate(x0)))
ISNATILISTKIND(n__cons(n__and(x0, x1), y1)) → AND(isNatKind(and(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → AND(isNatKind(cons(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__isNatIListKind(x0), y1)) → AND(isNatKind(isNatIListKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__s(x0), y1)) → AND(s(activate(x0)), y1)
ACTIVATE(n__and(n__isNatIListKind(x0), y1)) → AND(isNatIListKind(x0), y1)
ISNATKIND(n__s(n__nil)) → ISNATKIND(nil)
AND(tt, X) → ACTIVATE(X)
ISNATKIND(n__s(n__0)) → ISNATKIND(0)
ACTIVATE(n__and(x0, y1)) → AND(x0, y1)
ISNATKIND(n__s(n__zeros)) → ISNATKIND(zeros)
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ISNATILISTKIND(n__cons(n__isNat(x0), y1)) → AND(isNatKind(isNat(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATILISTKIND(n__cons(n__0, y1)) → AND(isNatKind(0), n__isNatIListKind(activate(y1)))

The TRS R consists of the following rules:

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

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

ACTIVATE(n__and(n__nil, y0)) → AND(n__nil, y0)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                              ↳ QDP

                    ↳ QDP

          ↳ QDP

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

ISNATKIND(n__s(n__isNat(x0))) → ISNATKIND(isNat(x0))
ACTIVATE(n__and(n__isNatKind(x0), y1)) → AND(isNatKind(x0), y1)
ISNATKIND(n__s(x0)) → ISNATKIND(x0)
ISNATKIND(n__s(n__cons(x0, x1))) → ISNATKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(n__isNatKind(x0), y1)) → AND(isNatKind(isNatKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__nil, y0)) → AND(n__nil, y0)
ACTIVATE(n__and(n__length(x0), y1)) → AND(length(activate(x0)), y1)
ACTIVATE(n__and(n__and(x0, x1), y1)) → AND(and(activate(x0), x1), y1)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ISNATILISTKIND(n__cons(x0, y1)) → AND(isNatKind(x0), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__isNat(x0), y1)) → AND(isNat(x0), y1)
ISNATILISTKIND(n__cons(n__length(x0), y1)) → AND(isNatKind(length(activate(x0))), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATKIND(n__s(n__isNatKind(x0))) → ISNATKIND(isNatKind(x0))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → AND(isNatKind(s(activate(x0))), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__zeros, y0)) → AND(cons(0, n__zeros), y0)
ISNATKIND(n__s(n__and(x0, x1))) → ISNATKIND(and(activate(x0), x1))
ISNATKIND(n__s(n__isNatIListKind(x0))) → ISNATKIND(isNatIListKind(x0))
ACTIVATE(n__and(n__0, y1)) → AND(0, y1)
ISNATILISTKIND(n__cons(n__nil, y1)) → AND(isNatKind(nil), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__zeros, y1)) → AND(isNatKind(zeros), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__cons(x0, x1), y1)) → AND(cons(activate(x0), x1), y1)
ISNATKIND(n__s(n__length(x0))) → ISNATKIND(length(activate(x0)))
ISNATKIND(n__s(n__s(x0))) → ISNATKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__and(x0, x1), y1)) → AND(isNatKind(and(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → AND(isNatKind(cons(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__isNatIListKind(x0), y1)) → AND(isNatKind(isNatIListKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__isNatIListKind(x0), y1)) → AND(isNatIListKind(x0), y1)
ACTIVATE(n__and(n__s(x0), y1)) → AND(s(activate(x0)), y1)
AND(tt, X) → ACTIVATE(X)
ISNATKIND(n__s(n__nil)) → ISNATKIND(nil)
ISNATKIND(n__s(n__0)) → ISNATKIND(0)
ACTIVATE(n__and(x0, y1)) → AND(x0, y1)
ISNATKIND(n__s(n__zeros)) → ISNATKIND(zeros)
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ISNATILISTKIND(n__cons(n__isNat(x0), y1)) → AND(isNatKind(isNat(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATILISTKIND(n__cons(n__0, y1)) → AND(isNatKind(0), n__isNatIListKind(activate(y1)))

The TRS R consists of the following rules:

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

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

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                              ↳ QDP

                    ↳ QDP

          ↳ QDP

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

ISNATKIND(n__s(n__isNat(x0))) → ISNATKIND(isNat(x0))
ACTIVATE(n__and(n__isNatKind(x0), y1)) → AND(isNatKind(x0), y1)
ISNATKIND(n__s(x0)) → ISNATKIND(x0)
ISNATKIND(n__s(n__cons(x0, x1))) → ISNATKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(n__isNatKind(x0), y1)) → AND(isNatKind(isNatKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__length(x0), y1)) → AND(length(activate(x0)), y1)
ACTIVATE(n__and(n__and(x0, x1), y1)) → AND(and(activate(x0), x1), y1)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ISNATILISTKIND(n__cons(x0, y1)) → AND(isNatKind(x0), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → AND(isNatKind(length(activate(x0))), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__isNat(x0), y1)) → AND(isNat(x0), y1)
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATKIND(n__s(n__isNatKind(x0))) → ISNATKIND(isNatKind(x0))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → AND(isNatKind(s(activate(x0))), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(n__and(x0, x1))) → ISNATKIND(and(activate(x0), x1))
ACTIVATE(n__and(n__zeros, y0)) → AND(cons(0, n__zeros), y0)
ISNATKIND(n__s(n__isNatIListKind(x0))) → ISNATKIND(isNatIListKind(x0))
ACTIVATE(n__and(n__0, y1)) → AND(0, y1)
ISNATILISTKIND(n__cons(n__zeros, y1)) → AND(isNatKind(zeros), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__nil, y1)) → AND(isNatKind(nil), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__cons(x0, x1), y1)) → AND(cons(activate(x0), x1), y1)
ISNATKIND(n__s(n__length(x0))) → ISNATKIND(length(activate(x0)))
ISNATKIND(n__s(n__s(x0))) → ISNATKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__and(x0, x1), y1)) → AND(isNatKind(and(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → AND(isNatKind(cons(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__isNatIListKind(x0), y1)) → AND(isNatKind(isNatIListKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__isNatIListKind(x0), y1)) → AND(isNatIListKind(x0), y1)
ACTIVATE(n__and(n__s(x0), y1)) → AND(s(activate(x0)), y1)
AND(tt, X) → ACTIVATE(X)
ISNATKIND(n__s(n__nil)) → ISNATKIND(nil)
ISNATKIND(n__s(n__0)) → ISNATKIND(0)
ACTIVATE(n__and(x0, y1)) → AND(x0, y1)
ISNATKIND(n__s(n__zeros)) → ISNATKIND(zeros)
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ISNATILISTKIND(n__cons(n__isNat(x0), y1)) → AND(isNatKind(isNat(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATILISTKIND(n__cons(n__0, y1)) → AND(isNatKind(0), n__isNatIListKind(activate(y1)))

The TRS R consists of the following rules:

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

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

ACTIVATE(n__and(n__0, y0)) → AND(n__0, y0)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                              ↳ QDP

                    ↳ QDP

          ↳ QDP

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

ISNATKIND(n__s(n__isNat(x0))) → ISNATKIND(isNat(x0))
ACTIVATE(n__and(n__isNatKind(x0), y1)) → AND(isNatKind(x0), y1)
ISNATKIND(n__s(x0)) → ISNATKIND(x0)
ISNATKIND(n__s(n__cons(x0, x1))) → ISNATKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(n__isNatKind(x0), y1)) → AND(isNatKind(isNatKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__length(x0), y1)) → AND(length(activate(x0)), y1)
ACTIVATE(n__and(n__and(x0, x1), y1)) → AND(and(activate(x0), x1), y1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ISNATILISTKIND(n__cons(x0, y1)) → AND(isNatKind(x0), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__isNat(x0), y1)) → AND(isNat(x0), y1)
ISNATILISTKIND(n__cons(n__length(x0), y1)) → AND(isNatKind(length(activate(x0))), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ACTIVATE(n__and(n__0, y0)) → AND(n__0, y0)
ISNATKIND(n__s(n__isNatKind(x0))) → ISNATKIND(isNatKind(x0))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → AND(isNatKind(s(activate(x0))), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__zeros, y0)) → AND(cons(0, n__zeros), y0)
ISNATKIND(n__s(n__and(x0, x1))) → ISNATKIND(and(activate(x0), x1))
ISNATKIND(n__s(n__isNatIListKind(x0))) → ISNATKIND(isNatIListKind(x0))
ISNATILISTKIND(n__cons(n__nil, y1)) → AND(isNatKind(nil), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__zeros, y1)) → AND(isNatKind(zeros), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__cons(x0, x1), y1)) → AND(cons(activate(x0), x1), y1)
ISNATKIND(n__s(n__s(x0))) → ISNATKIND(s(activate(x0)))
ISNATKIND(n__s(n__length(x0))) → ISNATKIND(length(activate(x0)))
ISNATILISTKIND(n__cons(n__and(x0, x1), y1)) → AND(isNatKind(and(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → AND(isNatKind(cons(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__isNatIListKind(x0), y1)) → AND(isNatKind(isNatIListKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__s(x0), y1)) → AND(s(activate(x0)), y1)
ACTIVATE(n__and(n__isNatIListKind(x0), y1)) → AND(isNatIListKind(x0), y1)
ISNATKIND(n__s(n__nil)) → ISNATKIND(nil)
AND(tt, X) → ACTIVATE(X)
ISNATKIND(n__s(n__0)) → ISNATKIND(0)
ACTIVATE(n__and(x0, y1)) → AND(x0, y1)
ISNATKIND(n__s(n__zeros)) → ISNATKIND(zeros)
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ISNATILISTKIND(n__cons(n__isNat(x0), y1)) → AND(isNatKind(isNat(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATILISTKIND(n__cons(n__0, y1)) → AND(isNatKind(0), n__isNatIListKind(activate(y1)))

The TRS R consists of the following rules:

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

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

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                              ↳ QDP

                    ↳ QDP

          ↳ QDP

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

ISNATKIND(n__s(n__isNat(x0))) → ISNATKIND(isNat(x0))
ACTIVATE(n__and(n__isNatKind(x0), y1)) → AND(isNatKind(x0), y1)
ISNATKIND(n__s(x0)) → ISNATKIND(x0)
ISNATKIND(n__s(n__cons(x0, x1))) → ISNATKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(n__isNatKind(x0), y1)) → AND(isNatKind(isNatKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__length(x0), y1)) → AND(length(activate(x0)), y1)
ACTIVATE(n__and(n__and(x0, x1), y1)) → AND(and(activate(x0), x1), y1)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ISNATILISTKIND(n__cons(x0, y1)) → AND(isNatKind(x0), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → AND(isNatKind(length(activate(x0))), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__isNat(x0), y1)) → AND(isNat(x0), y1)
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATKIND(n__s(n__isNatKind(x0))) → ISNATKIND(isNatKind(x0))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → AND(isNatKind(s(activate(x0))), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(n__and(x0, x1))) → ISNATKIND(and(activate(x0), x1))
ACTIVATE(n__and(n__zeros, y0)) → AND(cons(0, n__zeros), y0)
ISNATKIND(n__s(n__isNatIListKind(x0))) → ISNATKIND(isNatIListKind(x0))
ISNATILISTKIND(n__cons(n__zeros, y1)) → AND(isNatKind(zeros), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__nil, y1)) → AND(isNatKind(nil), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__cons(x0, x1), y1)) → AND(cons(activate(x0), x1), y1)
ISNATKIND(n__s(n__length(x0))) → ISNATKIND(length(activate(x0)))
ISNATKIND(n__s(n__s(x0))) → ISNATKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__and(x0, x1), y1)) → AND(isNatKind(and(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → AND(isNatKind(cons(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__isNatIListKind(x0), y1)) → AND(isNatKind(isNatIListKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__isNatIListKind(x0), y1)) → AND(isNatIListKind(x0), y1)
ACTIVATE(n__and(n__s(x0), y1)) → AND(s(activate(x0)), y1)
AND(tt, X) → ACTIVATE(X)
ISNATKIND(n__s(n__nil)) → ISNATKIND(nil)
ISNATKIND(n__s(n__0)) → ISNATKIND(0)
ACTIVATE(n__and(x0, y1)) → AND(x0, y1)
ISNATKIND(n__s(n__zeros)) → ISNATKIND(zeros)
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ISNATILISTKIND(n__cons(n__isNat(x0), y1)) → AND(isNatKind(isNat(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATILISTKIND(n__cons(n__0, y1)) → AND(isNatKind(0), n__isNatIListKind(activate(y1)))

The TRS R consists of the following rules:

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

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

ACTIVATE(n__and(n__zeros, y0)) → AND(cons(n__0, n__zeros), y0)
ACTIVATE(n__and(n__zeros, y0)) → AND(n__cons(0, n__zeros), y0)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                              ↳ QDP

                    ↳ QDP

          ↳ QDP

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

ISNATKIND(n__s(n__isNat(x0))) → ISNATKIND(isNat(x0))
ACTIVATE(n__and(n__isNatKind(x0), y1)) → AND(isNatKind(x0), y1)
ISNATKIND(n__s(x0)) → ISNATKIND(x0)
ISNATKIND(n__s(n__cons(x0, x1))) → ISNATKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(n__isNatKind(x0), y1)) → AND(isNatKind(isNatKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__length(x0), y1)) → AND(length(activate(x0)), y1)
ACTIVATE(n__and(n__and(x0, x1), y1)) → AND(and(activate(x0), x1), y1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ISNATILISTKIND(n__cons(x0, y1)) → AND(isNatKind(x0), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__zeros, y0)) → AND(n__cons(0, n__zeros), y0)
ACTIVATE(n__and(n__isNat(x0), y1)) → AND(isNat(x0), y1)
ISNATILISTKIND(n__cons(n__length(x0), y1)) → AND(isNatKind(length(activate(x0))), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATKIND(n__s(n__isNatKind(x0))) → ISNATKIND(isNatKind(x0))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → AND(isNatKind(s(activate(x0))), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(n__and(x0, x1))) → ISNATKIND(and(activate(x0), x1))
ISNATKIND(n__s(n__isNatIListKind(x0))) → ISNATKIND(isNatIListKind(x0))
ISNATILISTKIND(n__cons(n__nil, y1)) → AND(isNatKind(nil), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__zeros, y1)) → AND(isNatKind(zeros), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__cons(x0, x1), y1)) → AND(cons(activate(x0), x1), y1)
ISNATKIND(n__s(n__s(x0))) → ISNATKIND(s(activate(x0)))
ISNATKIND(n__s(n__length(x0))) → ISNATKIND(length(activate(x0)))
ISNATILISTKIND(n__cons(n__and(x0, x1), y1)) → AND(isNatKind(and(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → AND(isNatKind(cons(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__isNatIListKind(x0), y1)) → AND(isNatKind(isNatIListKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__s(x0), y1)) → AND(s(activate(x0)), y1)
ACTIVATE(n__and(n__isNatIListKind(x0), y1)) → AND(isNatIListKind(x0), y1)
ISNATKIND(n__s(n__nil)) → ISNATKIND(nil)
AND(tt, X) → ACTIVATE(X)
ISNATKIND(n__s(n__0)) → ISNATKIND(0)
ACTIVATE(n__and(x0, y1)) → AND(x0, y1)
ISNATKIND(n__s(n__zeros)) → ISNATKIND(zeros)
ACTIVATE(n__and(n__zeros, y0)) → AND(cons(n__0, n__zeros), y0)
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ISNATILISTKIND(n__cons(n__isNat(x0), y1)) → AND(isNatKind(isNat(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATILISTKIND(n__cons(n__0, y1)) → AND(isNatKind(0), n__isNatIListKind(activate(y1)))

The TRS R consists of the following rules:

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

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

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ 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

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

ISNATKIND(n__s(n__isNat(x0))) → ISNATKIND(isNat(x0))
ACTIVATE(n__and(n__isNatKind(x0), y1)) → AND(isNatKind(x0), y1)
ISNATKIND(n__s(x0)) → ISNATKIND(x0)
ISNATKIND(n__s(n__cons(x0, x1))) → ISNATKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(n__isNatKind(x0), y1)) → AND(isNatKind(isNatKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__length(x0), y1)) → AND(length(activate(x0)), y1)
ACTIVATE(n__and(n__and(x0, x1), y1)) → AND(and(activate(x0), x1), y1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ISNATILISTKIND(n__cons(x0, y1)) → AND(isNatKind(x0), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → AND(isNatKind(length(activate(x0))), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__isNat(x0), y1)) → AND(isNat(x0), y1)
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATKIND(n__s(n__isNatKind(x0))) → ISNATKIND(isNatKind(x0))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → AND(isNatKind(s(activate(x0))), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(n__and(x0, x1))) → ISNATKIND(and(activate(x0), x1))
ISNATKIND(n__s(n__isNatIListKind(x0))) → ISNATKIND(isNatIListKind(x0))
ISNATILISTKIND(n__cons(n__zeros, y1)) → AND(isNatKind(zeros), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__nil, y1)) → AND(isNatKind(nil), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__cons(x0, x1), y1)) → AND(cons(activate(x0), x1), y1)
ISNATKIND(n__s(n__s(x0))) → ISNATKIND(s(activate(x0)))
ISNATKIND(n__s(n__length(x0))) → ISNATKIND(length(activate(x0)))
ISNATILISTKIND(n__cons(n__and(x0, x1), y1)) → AND(isNatKind(and(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → AND(isNatKind(cons(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__isNatIListKind(x0), y1)) → AND(isNatKind(isNatIListKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__s(x0), y1)) → AND(s(activate(x0)), y1)
ACTIVATE(n__and(n__isNatIListKind(x0), y1)) → AND(isNatIListKind(x0), y1)
ISNATKIND(n__s(n__nil)) → ISNATKIND(nil)
AND(tt, X) → ACTIVATE(X)
ISNATKIND(n__s(n__0)) → ISNATKIND(0)
ACTIVATE(n__and(x0, y1)) → AND(x0, y1)
ISNATKIND(n__s(n__zeros)) → ISNATKIND(zeros)
ACTIVATE(n__and(n__zeros, y0)) → AND(cons(n__0, n__zeros), y0)
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ISNATILISTKIND(n__cons(n__isNat(x0), y1)) → AND(isNatKind(isNat(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATILISTKIND(n__cons(n__0, y1)) → AND(isNatKind(0), n__isNatIListKind(activate(y1)))

The TRS R consists of the following rules:

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

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

ACTIVATE(n__and(n__zeros, y0)) → AND(n__cons(n__0, n__zeros), y0)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ 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

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

ISNATKIND(n__s(n__isNat(x0))) → ISNATKIND(isNat(x0))
ACTIVATE(n__and(n__isNatKind(x0), y1)) → AND(isNatKind(x0), y1)
ISNATKIND(n__s(x0)) → ISNATKIND(x0)
ISNATKIND(n__s(n__cons(x0, x1))) → ISNATKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(n__isNatKind(x0), y1)) → AND(isNatKind(isNatKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__length(x0), y1)) → AND(length(activate(x0)), y1)
ACTIVATE(n__and(n__and(x0, x1), y1)) → AND(and(activate(x0), x1), y1)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ISNATILISTKIND(n__cons(x0, y1)) → AND(isNatKind(x0), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__isNat(x0), y1)) → AND(isNat(x0), y1)
ISNATILISTKIND(n__cons(n__length(x0), y1)) → AND(isNatKind(length(activate(x0))), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ACTIVATE(n__and(n__zeros, y0)) → AND(n__cons(n__0, n__zeros), y0)
ISNATKIND(n__s(n__isNatKind(x0))) → ISNATKIND(isNatKind(x0))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → AND(isNatKind(s(activate(x0))), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(n__and(x0, x1))) → ISNATKIND(and(activate(x0), x1))
ISNATKIND(n__s(n__isNatIListKind(x0))) → ISNATKIND(isNatIListKind(x0))
ISNATILISTKIND(n__cons(n__nil, y1)) → AND(isNatKind(nil), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__zeros, y1)) → AND(isNatKind(zeros), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__cons(x0, x1), y1)) → AND(cons(activate(x0), x1), y1)
ISNATKIND(n__s(n__length(x0))) → ISNATKIND(length(activate(x0)))
ISNATKIND(n__s(n__s(x0))) → ISNATKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__and(x0, x1), y1)) → AND(isNatKind(and(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → AND(isNatKind(cons(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__isNatIListKind(x0), y1)) → AND(isNatKind(isNatIListKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__isNatIListKind(x0), y1)) → AND(isNatIListKind(x0), y1)
ACTIVATE(n__and(n__s(x0), y1)) → AND(s(activate(x0)), y1)
AND(tt, X) → ACTIVATE(X)
ISNATKIND(n__s(n__nil)) → ISNATKIND(nil)
ISNATKIND(n__s(n__0)) → ISNATKIND(0)
ACTIVATE(n__and(x0, y1)) → AND(x0, y1)
ISNATKIND(n__s(n__zeros)) → ISNATKIND(zeros)
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ISNATILISTKIND(n__cons(n__isNat(x0), y1)) → AND(isNatKind(isNat(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATILISTKIND(n__cons(n__0, y1)) → AND(isNatKind(0), n__isNatIListKind(activate(y1)))

The TRS R consists of the following rules:

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

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

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ 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

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

ISNATKIND(n__s(n__isNat(x0))) → ISNATKIND(isNat(x0))
ACTIVATE(n__and(n__isNatKind(x0), y1)) → AND(isNatKind(x0), y1)
ISNATKIND(n__s(x0)) → ISNATKIND(x0)
ISNATKIND(n__s(n__cons(x0, x1))) → ISNATKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(n__isNatKind(x0), y1)) → AND(isNatKind(isNatKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__length(x0), y1)) → AND(length(activate(x0)), y1)
ACTIVATE(n__and(n__and(x0, x1), y1)) → AND(and(activate(x0), x1), y1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ISNATILISTKIND(n__cons(x0, y1)) → AND(isNatKind(x0), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → AND(isNatKind(length(activate(x0))), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__isNat(x0), y1)) → AND(isNat(x0), y1)
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATKIND(n__s(n__isNatKind(x0))) → ISNATKIND(isNatKind(x0))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → AND(isNatKind(s(activate(x0))), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(n__and(x0, x1))) → ISNATKIND(and(activate(x0), x1))
ISNATKIND(n__s(n__isNatIListKind(x0))) → ISNATKIND(isNatIListKind(x0))
ISNATILISTKIND(n__cons(n__zeros, y1)) → AND(isNatKind(zeros), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__nil, y1)) → AND(isNatKind(nil), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__cons(x0, x1), y1)) → AND(cons(activate(x0), x1), y1)
ISNATKIND(n__s(n__s(x0))) → ISNATKIND(s(activate(x0)))
ISNATKIND(n__s(n__length(x0))) → ISNATKIND(length(activate(x0)))
ISNATILISTKIND(n__cons(n__and(x0, x1), y1)) → AND(isNatKind(and(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → AND(isNatKind(cons(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__isNatIListKind(x0), y1)) → AND(isNatKind(isNatIListKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__s(x0), y1)) → AND(s(activate(x0)), y1)
ACTIVATE(n__and(n__isNatIListKind(x0), y1)) → AND(isNatIListKind(x0), y1)
ISNATKIND(n__s(n__nil)) → ISNATKIND(nil)
AND(tt, X) → ACTIVATE(X)
ISNATKIND(n__s(n__0)) → ISNATKIND(0)
ACTIVATE(n__and(x0, y1)) → AND(x0, y1)
ISNATKIND(n__s(n__zeros)) → ISNATKIND(zeros)
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ISNATILISTKIND(n__cons(n__isNat(x0), y1)) → AND(isNatKind(isNat(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATILISTKIND(n__cons(n__0, y1)) → AND(isNatKind(0), n__isNatIListKind(activate(y1)))

The TRS R consists of the following rules:

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

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

ISNATKIND(n__s(n__zeros)) → ISNATKIND(cons(0, n__zeros))
ISNATKIND(n__s(n__zeros)) → ISNATKIND(n__zeros)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ 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

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

ISNATKIND(n__s(n__isNat(x0))) → ISNATKIND(isNat(x0))
ISNATKIND(n__s(n__zeros)) → ISNATKIND(cons(0, n__zeros))
ACTIVATE(n__and(n__isNatKind(x0), y1)) → AND(isNatKind(x0), y1)
ISNATKIND(n__s(x0)) → ISNATKIND(x0)
ISNATKIND(n__s(n__cons(x0, x1))) → ISNATKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(n__isNatKind(x0), y1)) → AND(isNatKind(isNatKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__length(x0), y1)) → AND(length(activate(x0)), y1)
ISNATKIND(n__s(n__zeros)) → ISNATKIND(n__zeros)
ACTIVATE(n__and(n__and(x0, x1), y1)) → AND(and(activate(x0), x1), y1)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ISNATILISTKIND(n__cons(x0, y1)) → AND(isNatKind(x0), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__isNat(x0), y1)) → AND(isNat(x0), y1)
ISNATILISTKIND(n__cons(n__length(x0), y1)) → AND(isNatKind(length(activate(x0))), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATKIND(n__s(n__isNatKind(x0))) → ISNATKIND(isNatKind(x0))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → AND(isNatKind(s(activate(x0))), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(n__and(x0, x1))) → ISNATKIND(and(activate(x0), x1))
ISNATKIND(n__s(n__isNatIListKind(x0))) → ISNATKIND(isNatIListKind(x0))
ISNATILISTKIND(n__cons(n__nil, y1)) → AND(isNatKind(nil), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__zeros, y1)) → AND(isNatKind(zeros), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__cons(x0, x1), y1)) → AND(cons(activate(x0), x1), y1)
ISNATKIND(n__s(n__length(x0))) → ISNATKIND(length(activate(x0)))
ISNATKIND(n__s(n__s(x0))) → ISNATKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__and(x0, x1), y1)) → AND(isNatKind(and(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → AND(isNatKind(cons(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__isNatIListKind(x0), y1)) → AND(isNatKind(isNatIListKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__isNatIListKind(x0), y1)) → AND(isNatIListKind(x0), y1)
ACTIVATE(n__and(n__s(x0), y1)) → AND(s(activate(x0)), y1)
AND(tt, X) → ACTIVATE(X)
ISNATKIND(n__s(n__nil)) → ISNATKIND(nil)
ISNATKIND(n__s(n__0)) → ISNATKIND(0)
ACTIVATE(n__and(x0, y1)) → AND(x0, y1)
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ISNATILISTKIND(n__cons(n__isNat(x0), y1)) → AND(isNatKind(isNat(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATILISTKIND(n__cons(n__0, y1)) → AND(isNatKind(0), n__isNatIListKind(activate(y1)))

The TRS R consists of the following rules:

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

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

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ 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

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

ISNATKIND(n__s(n__isNat(x0))) → ISNATKIND(isNat(x0))
ISNATKIND(n__s(n__zeros)) → ISNATKIND(cons(0, n__zeros))
ACTIVATE(n__and(n__isNatKind(x0), y1)) → AND(isNatKind(x0), y1)
ISNATKIND(n__s(x0)) → ISNATKIND(x0)
ISNATKIND(n__s(n__cons(x0, x1))) → ISNATKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(n__isNatKind(x0), y1)) → AND(isNatKind(isNatKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__length(x0), y1)) → AND(length(activate(x0)), y1)
ACTIVATE(n__and(n__and(x0, x1), y1)) → AND(and(activate(x0), x1), y1)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ISNATILISTKIND(n__cons(x0, y1)) → AND(isNatKind(x0), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → AND(isNatKind(length(activate(x0))), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__isNat(x0), y1)) → AND(isNat(x0), y1)
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATKIND(n__s(n__isNatKind(x0))) → ISNATKIND(isNatKind(x0))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → AND(isNatKind(s(activate(x0))), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(n__and(x0, x1))) → ISNATKIND(and(activate(x0), x1))
ISNATKIND(n__s(n__isNatIListKind(x0))) → ISNATKIND(isNatIListKind(x0))
ISNATILISTKIND(n__cons(n__zeros, y1)) → AND(isNatKind(zeros), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__nil, y1)) → AND(isNatKind(nil), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__cons(x0, x1), y1)) → AND(cons(activate(x0), x1), y1)
ISNATKIND(n__s(n__length(x0))) → ISNATKIND(length(activate(x0)))
ISNATKIND(n__s(n__s(x0))) → ISNATKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__and(x0, x1), y1)) → AND(isNatKind(and(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → AND(isNatKind(cons(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__isNatIListKind(x0), y1)) → AND(isNatKind(isNatIListKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__isNatIListKind(x0), y1)) → AND(isNatIListKind(x0), y1)
ACTIVATE(n__and(n__s(x0), y1)) → AND(s(activate(x0)), y1)
AND(tt, X) → ACTIVATE(X)
ISNATKIND(n__s(n__nil)) → ISNATKIND(nil)
ISNATKIND(n__s(n__0)) → ISNATKIND(0)
ACTIVATE(n__and(x0, y1)) → AND(x0, y1)
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ISNATILISTKIND(n__cons(n__isNat(x0), y1)) → AND(isNatKind(isNat(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATILISTKIND(n__cons(n__0, y1)) → AND(isNatKind(0), n__isNatIListKind(activate(y1)))

The TRS R consists of the following rules:

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

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

ISNATKIND(n__s(n__nil)) → ISNATKIND(n__nil)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ 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

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

ISNATKIND(n__s(n__isNat(x0))) → ISNATKIND(isNat(x0))
ACTIVATE(n__and(n__isNatKind(x0), y1)) → AND(isNatKind(x0), y1)
ISNATKIND(n__s(n__zeros)) → ISNATKIND(cons(0, n__zeros))
ISNATKIND(n__s(x0)) → ISNATKIND(x0)
ISNATKIND(n__s(n__cons(x0, x1))) → ISNATKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(n__isNatKind(x0), y1)) → AND(isNatKind(isNatKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__length(x0), y1)) → AND(length(activate(x0)), y1)
ACTIVATE(n__and(n__and(x0, x1), y1)) → AND(and(activate(x0), x1), y1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ISNATILISTKIND(n__cons(x0, y1)) → AND(isNatKind(x0), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__isNat(x0), y1)) → AND(isNat(x0), y1)
ISNATILISTKIND(n__cons(n__length(x0), y1)) → AND(isNatKind(length(activate(x0))), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATKIND(n__s(n__nil)) → ISNATKIND(n__nil)
ISNATKIND(n__s(n__isNatKind(x0))) → ISNATKIND(isNatKind(x0))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → AND(isNatKind(s(activate(x0))), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(n__and(x0, x1))) → ISNATKIND(and(activate(x0), x1))
ISNATKIND(n__s(n__isNatIListKind(x0))) → ISNATKIND(isNatIListKind(x0))
ISNATILISTKIND(n__cons(n__nil, y1)) → AND(isNatKind(nil), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__zeros, y1)) → AND(isNatKind(zeros), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__cons(x0, x1), y1)) → AND(cons(activate(x0), x1), y1)
ISNATKIND(n__s(n__s(x0))) → ISNATKIND(s(activate(x0)))
ISNATKIND(n__s(n__length(x0))) → ISNATKIND(length(activate(x0)))
ISNATILISTKIND(n__cons(n__and(x0, x1), y1)) → AND(isNatKind(and(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → AND(isNatKind(cons(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__isNatIListKind(x0), y1)) → AND(isNatKind(isNatIListKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__s(x0), y1)) → AND(s(activate(x0)), y1)
ACTIVATE(n__and(n__isNatIListKind(x0), y1)) → AND(isNatIListKind(x0), y1)
AND(tt, X) → ACTIVATE(X)
ISNATKIND(n__s(n__0)) → ISNATKIND(0)
ACTIVATE(n__and(x0, y1)) → AND(x0, y1)
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ISNATILISTKIND(n__cons(n__isNat(x0), y1)) → AND(isNatKind(isNat(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATILISTKIND(n__cons(n__0, y1)) → AND(isNatKind(0), n__isNatIListKind(activate(y1)))

The TRS R consists of the following rules:

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

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

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ 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

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

ISNATKIND(n__s(n__isNat(x0))) → ISNATKIND(isNat(x0))
ACTIVATE(n__and(n__isNatKind(x0), y1)) → AND(isNatKind(x0), y1)
ISNATKIND(n__s(n__zeros)) → ISNATKIND(cons(0, n__zeros))
ISNATKIND(n__s(x0)) → ISNATKIND(x0)
ISNATKIND(n__s(n__cons(x0, x1))) → ISNATKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(n__isNatKind(x0), y1)) → AND(isNatKind(isNatKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__length(x0), y1)) → AND(length(activate(x0)), y1)
ACTIVATE(n__and(n__and(x0, x1), y1)) → AND(and(activate(x0), x1), y1)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ISNATILISTKIND(n__cons(x0, y1)) → AND(isNatKind(x0), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → AND(isNatKind(length(activate(x0))), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__isNat(x0), y1)) → AND(isNat(x0), y1)
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATKIND(n__s(n__isNatKind(x0))) → ISNATKIND(isNatKind(x0))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → AND(isNatKind(s(activate(x0))), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(n__and(x0, x1))) → ISNATKIND(and(activate(x0), x1))
ISNATKIND(n__s(n__isNatIListKind(x0))) → ISNATKIND(isNatIListKind(x0))
ISNATILISTKIND(n__cons(n__zeros, y1)) → AND(isNatKind(zeros), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__nil, y1)) → AND(isNatKind(nil), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__cons(x0, x1), y1)) → AND(cons(activate(x0), x1), y1)
ISNATKIND(n__s(n__length(x0))) → ISNATKIND(length(activate(x0)))
ISNATKIND(n__s(n__s(x0))) → ISNATKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__and(x0, x1), y1)) → AND(isNatKind(and(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → AND(isNatKind(cons(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__isNatIListKind(x0), y1)) → AND(isNatKind(isNatIListKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__isNatIListKind(x0), y1)) → AND(isNatIListKind(x0), y1)
ACTIVATE(n__and(n__s(x0), y1)) → AND(s(activate(x0)), y1)
AND(tt, X) → ACTIVATE(X)
ISNATKIND(n__s(n__0)) → ISNATKIND(0)
ACTIVATE(n__and(x0, y1)) → AND(x0, y1)
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ISNATILISTKIND(n__cons(n__isNat(x0), y1)) → AND(isNatKind(isNat(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATILISTKIND(n__cons(n__0, y1)) → AND(isNatKind(0), n__isNatIListKind(activate(y1)))

The TRS R consists of the following rules:

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

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

ISNATKIND(n__s(n__0)) → ISNATKIND(n__0)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ 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

                                                                                                                    ↳ DependencyGraphProof

                              ↳ QDP

                    ↳ QDP

          ↳ QDP

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

ISNATKIND(n__s(n__isNat(x0))) → ISNATKIND(isNat(x0))
ISNATKIND(n__s(n__zeros)) → ISNATKIND(cons(0, n__zeros))
ACTIVATE(n__and(n__isNatKind(x0), y1)) → AND(isNatKind(x0), y1)
ISNATKIND(n__s(x0)) → ISNATKIND(x0)
ISNATKIND(n__s(n__cons(x0, x1))) → ISNATKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(n__isNatKind(x0), y1)) → AND(isNatKind(isNatKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__length(x0), y1)) → AND(length(activate(x0)), y1)
ACTIVATE(n__and(n__and(x0, x1), y1)) → AND(and(activate(x0), x1), y1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ISNATILISTKIND(n__cons(x0, y1)) → AND(isNatKind(x0), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__isNat(x0), y1)) → AND(isNat(x0), y1)
ISNATILISTKIND(n__cons(n__length(x0), y1)) → AND(isNatKind(length(activate(x0))), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATKIND(n__s(n__isNatKind(x0))) → ISNATKIND(isNatKind(x0))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → AND(isNatKind(s(activate(x0))), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(n__and(x0, x1))) → ISNATKIND(and(activate(x0), x1))
ISNATKIND(n__s(n__isNatIListKind(x0))) → ISNATKIND(isNatIListKind(x0))
ISNATILISTKIND(n__cons(n__nil, y1)) → AND(isNatKind(nil), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__zeros, y1)) → AND(isNatKind(zeros), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(n__0)) → ISNATKIND(n__0)
ACTIVATE(n__and(n__cons(x0, x1), y1)) → AND(cons(activate(x0), x1), y1)
ISNATKIND(n__s(n__s(x0))) → ISNATKIND(s(activate(x0)))
ISNATKIND(n__s(n__length(x0))) → ISNATKIND(length(activate(x0)))
ISNATILISTKIND(n__cons(n__and(x0, x1), y1)) → AND(isNatKind(and(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → AND(isNatKind(cons(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__isNatIListKind(x0), y1)) → AND(isNatKind(isNatIListKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__s(x0), y1)) → AND(s(activate(x0)), y1)
ACTIVATE(n__and(n__isNatIListKind(x0), y1)) → AND(isNatIListKind(x0), y1)
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(x0, y1)) → AND(x0, y1)
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ISNATILISTKIND(n__cons(n__isNat(x0), y1)) → AND(isNatKind(isNat(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATILISTKIND(n__cons(n__0, y1)) → AND(isNatKind(0), n__isNatIListKind(activate(y1)))

The TRS R consists of the following rules:

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

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

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ 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

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                              ↳ QDP

                    ↳ QDP

          ↳ QDP

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

ISNATKIND(n__s(n__isNat(x0))) → ISNATKIND(isNat(x0))
ISNATKIND(n__s(n__zeros)) → ISNATKIND(cons(0, n__zeros))
ACTIVATE(n__and(n__isNatKind(x0), y1)) → AND(isNatKind(x0), y1)
ISNATKIND(n__s(x0)) → ISNATKIND(x0)
ISNATKIND(n__s(n__cons(x0, x1))) → ISNATKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(n__isNatKind(x0), y1)) → AND(isNatKind(isNatKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__length(x0), y1)) → AND(length(activate(x0)), y1)
ACTIVATE(n__and(n__and(x0, x1), y1)) → AND(and(activate(x0), x1), y1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ISNATILISTKIND(n__cons(x0, y1)) → AND(isNatKind(x0), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → AND(isNatKind(length(activate(x0))), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__isNat(x0), y1)) → AND(isNat(x0), y1)
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATKIND(n__s(n__isNatKind(x0))) → ISNATKIND(isNatKind(x0))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → AND(isNatKind(s(activate(x0))), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(n__and(x0, x1))) → ISNATKIND(and(activate(x0), x1))
ISNATKIND(n__s(n__isNatIListKind(x0))) → ISNATKIND(isNatIListKind(x0))
ISNATILISTKIND(n__cons(n__zeros, y1)) → AND(isNatKind(zeros), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__nil, y1)) → AND(isNatKind(nil), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__cons(x0, x1), y1)) → AND(cons(activate(x0), x1), y1)
ISNATKIND(n__s(n__s(x0))) → ISNATKIND(s(activate(x0)))
ISNATKIND(n__s(n__length(x0))) → ISNATKIND(length(activate(x0)))
ISNATILISTKIND(n__cons(n__and(x0, x1), y1)) → AND(isNatKind(and(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → AND(isNatKind(cons(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__isNatIListKind(x0), y1)) → AND(isNatKind(isNatIListKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__s(x0), y1)) → AND(s(activate(x0)), y1)
ACTIVATE(n__and(n__isNatIListKind(x0), y1)) → AND(isNatIListKind(x0), y1)
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(x0, y1)) → AND(x0, y1)
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ISNATILISTKIND(n__cons(n__isNat(x0), y1)) → AND(isNatKind(isNat(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATILISTKIND(n__cons(n__0, y1)) → AND(isNatKind(0), n__isNatIListKind(activate(y1)))

The TRS R consists of the following rules:

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

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

ISNATKIND(n__s(n__zeros)) → ISNATKIND(cons(n__0, n__zeros))
ISNATKIND(n__s(n__zeros)) → ISNATKIND(n__cons(0, n__zeros))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ 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

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                              ↳ QDP

                    ↳ QDP

          ↳ QDP

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

ISNATKIND(n__s(n__isNat(x0))) → ISNATKIND(isNat(x0))
ACTIVATE(n__and(n__isNatKind(x0), y1)) → AND(isNatKind(x0), y1)
ISNATKIND(n__s(x0)) → ISNATKIND(x0)
ISNATKIND(n__s(n__cons(x0, x1))) → ISNATKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(n__isNatKind(x0), y1)) → AND(isNatKind(isNatKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__length(x0), y1)) → AND(length(activate(x0)), y1)
ACTIVATE(n__and(n__and(x0, x1), y1)) → AND(and(activate(x0), x1), y1)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ISNATILISTKIND(n__cons(x0, y1)) → AND(isNatKind(x0), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__isNat(x0), y1)) → AND(isNat(x0), y1)
ISNATILISTKIND(n__cons(n__length(x0), y1)) → AND(isNatKind(length(activate(x0))), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATKIND(n__s(n__isNatKind(x0))) → ISNATKIND(isNatKind(x0))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → AND(isNatKind(s(activate(x0))), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(n__and(x0, x1))) → ISNATKIND(and(activate(x0), x1))
ISNATKIND(n__s(n__isNatIListKind(x0))) → ISNATKIND(isNatIListKind(x0))
ISNATILISTKIND(n__cons(n__nil, y1)) → AND(isNatKind(nil), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__zeros, y1)) → AND(isNatKind(zeros), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__cons(x0, x1), y1)) → AND(cons(activate(x0), x1), y1)
ISNATKIND(n__s(n__length(x0))) → ISNATKIND(length(activate(x0)))
ISNATKIND(n__s(n__s(x0))) → ISNATKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__and(x0, x1), y1)) → AND(isNatKind(and(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → AND(isNatKind(cons(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__isNatIListKind(x0), y1)) → AND(isNatKind(isNatIListKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__isNatIListKind(x0), y1)) → AND(isNatIListKind(x0), y1)
ACTIVATE(n__and(n__s(x0), y1)) → AND(s(activate(x0)), y1)
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(x0, y1)) → AND(x0, y1)
ISNATKIND(n__s(n__zeros)) → ISNATKIND(cons(n__0, n__zeros))
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ISNATILISTKIND(n__cons(n__isNat(x0), y1)) → AND(isNatKind(isNat(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATKIND(n__s(n__zeros)) → ISNATKIND(n__cons(0, n__zeros))
ISNATILISTKIND(n__cons(n__0, y1)) → AND(isNatKind(0), n__isNatIListKind(activate(y1)))

The TRS R consists of the following rules:

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

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

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ 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

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                              ↳ QDP

                    ↳ QDP

          ↳ QDP

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

ISNATKIND(n__s(n__isNat(x0))) → ISNATKIND(isNat(x0))
ACTIVATE(n__and(n__isNatKind(x0), y1)) → AND(isNatKind(x0), y1)
ISNATKIND(n__s(x0)) → ISNATKIND(x0)
ISNATKIND(n__s(n__cons(x0, x1))) → ISNATKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(n__isNatKind(x0), y1)) → AND(isNatKind(isNatKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__length(x0), y1)) → AND(length(activate(x0)), y1)
ACTIVATE(n__and(n__and(x0, x1), y1)) → AND(and(activate(x0), x1), y1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ISNATILISTKIND(n__cons(x0, y1)) → AND(isNatKind(x0), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → AND(isNatKind(length(activate(x0))), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__isNat(x0), y1)) → AND(isNat(x0), y1)
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATKIND(n__s(n__isNatKind(x0))) → ISNATKIND(isNatKind(x0))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → AND(isNatKind(s(activate(x0))), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(n__and(x0, x1))) → ISNATKIND(and(activate(x0), x1))
ISNATKIND(n__s(n__isNatIListKind(x0))) → ISNATKIND(isNatIListKind(x0))
ISNATILISTKIND(n__cons(n__zeros, y1)) → AND(isNatKind(zeros), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__nil, y1)) → AND(isNatKind(nil), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__cons(x0, x1), y1)) → AND(cons(activate(x0), x1), y1)
ISNATKIND(n__s(n__s(x0))) → ISNATKIND(s(activate(x0)))
ISNATKIND(n__s(n__length(x0))) → ISNATKIND(length(activate(x0)))
ISNATILISTKIND(n__cons(n__and(x0, x1), y1)) → AND(isNatKind(and(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → AND(isNatKind(cons(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__isNatIListKind(x0), y1)) → AND(isNatKind(isNatIListKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__s(x0), y1)) → AND(s(activate(x0)), y1)
ACTIVATE(n__and(n__isNatIListKind(x0), y1)) → AND(isNatIListKind(x0), y1)
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(x0, y1)) → AND(x0, y1)
ISNATKIND(n__s(n__zeros)) → ISNATKIND(cons(n__0, n__zeros))
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ISNATILISTKIND(n__cons(n__isNat(x0), y1)) → AND(isNatKind(isNat(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATILISTKIND(n__cons(n__0, y1)) → AND(isNatKind(0), n__isNatIListKind(activate(y1)))

The TRS R consists of the following rules:

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

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

ISNATKIND(n__s(n__zeros)) → ISNATKIND(n__cons(n__0, n__zeros))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ 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

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                              ↳ QDP

                    ↳ QDP

          ↳ QDP

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

ISNATKIND(n__s(n__isNat(x0))) → ISNATKIND(isNat(x0))
ACTIVATE(n__and(n__isNatKind(x0), y1)) → AND(isNatKind(x0), y1)
ISNATKIND(n__s(x0)) → ISNATKIND(x0)
ISNATKIND(n__s(n__cons(x0, x1))) → ISNATKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(n__isNatKind(x0), y1)) → AND(isNatKind(isNatKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__length(x0), y1)) → AND(length(activate(x0)), y1)
ACTIVATE(n__and(n__and(x0, x1), y1)) → AND(and(activate(x0), x1), y1)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ISNATILISTKIND(n__cons(x0, y1)) → AND(isNatKind(x0), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__isNat(x0), y1)) → AND(isNat(x0), y1)
ISNATILISTKIND(n__cons(n__length(x0), y1)) → AND(isNatKind(length(activate(x0))), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATKIND(n__s(n__isNatKind(x0))) → ISNATKIND(isNatKind(x0))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → AND(isNatKind(s(activate(x0))), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(n__zeros)) → ISNATKIND(n__cons(n__0, n__zeros))
ISNATKIND(n__s(n__and(x0, x1))) → ISNATKIND(and(activate(x0), x1))
ISNATKIND(n__s(n__isNatIListKind(x0))) → ISNATKIND(isNatIListKind(x0))
ISNATILISTKIND(n__cons(n__nil, y1)) → AND(isNatKind(nil), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__zeros, y1)) → AND(isNatKind(zeros), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__cons(x0, x1), y1)) → AND(cons(activate(x0), x1), y1)
ISNATKIND(n__s(n__length(x0))) → ISNATKIND(length(activate(x0)))
ISNATKIND(n__s(n__s(x0))) → ISNATKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__and(x0, x1), y1)) → AND(isNatKind(and(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → AND(isNatKind(cons(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__isNatIListKind(x0), y1)) → AND(isNatKind(isNatIListKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__isNatIListKind(x0), y1)) → AND(isNatIListKind(x0), y1)
ACTIVATE(n__and(n__s(x0), y1)) → AND(s(activate(x0)), y1)
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(x0, y1)) → AND(x0, y1)
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ISNATILISTKIND(n__cons(n__isNat(x0), y1)) → AND(isNatKind(isNat(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATILISTKIND(n__cons(n__0, y1)) → AND(isNatKind(0), n__isNatIListKind(activate(y1)))

The TRS R consists of the following rules:

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

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

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ 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

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ QDPOrderProof

                              ↳ QDP

                    ↳ QDP

          ↳ QDP

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

ISNATKIND(n__s(n__isNat(x0))) → ISNATKIND(isNat(x0))
ACTIVATE(n__and(n__isNatKind(x0), y1)) → AND(isNatKind(x0), y1)
ISNATKIND(n__s(x0)) → ISNATKIND(x0)
ISNATKIND(n__s(n__cons(x0, x1))) → ISNATKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(n__isNatKind(x0), y1)) → AND(isNatKind(isNatKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__length(x0), y1)) → AND(length(activate(x0)), y1)
ACTIVATE(n__and(n__and(x0, x1), y1)) → AND(and(activate(x0), x1), y1)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ISNATILISTKIND(n__cons(x0, y1)) → AND(isNatKind(x0), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → AND(isNatKind(length(activate(x0))), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__isNat(x0), y1)) → AND(isNat(x0), y1)
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATKIND(n__s(n__isNatKind(x0))) → ISNATKIND(isNatKind(x0))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → AND(isNatKind(s(activate(x0))), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(n__and(x0, x1))) → ISNATKIND(and(activate(x0), x1))
ISNATKIND(n__s(n__isNatIListKind(x0))) → ISNATKIND(isNatIListKind(x0))
ISNATILISTKIND(n__cons(n__nil, y1)) → AND(isNatKind(nil), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__zeros, y1)) → AND(isNatKind(zeros), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__cons(x0, x1), y1)) → AND(cons(activate(x0), x1), y1)
ISNATKIND(n__s(n__length(x0))) → ISNATKIND(length(activate(x0)))
ISNATKIND(n__s(n__s(x0))) → ISNATKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__and(x0, x1), y1)) → AND(isNatKind(and(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → AND(isNatKind(cons(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__isNatIListKind(x0), y1)) → AND(isNatKind(isNatIListKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__isNatIListKind(x0), y1)) → AND(isNatIListKind(x0), y1)
ACTIVATE(n__and(n__s(x0), y1)) → AND(s(activate(x0)), y1)
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(x0, y1)) → AND(x0, y1)
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ISNATILISTKIND(n__cons(n__isNat(x0), y1)) → AND(isNatKind(isNat(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATILISTKIND(n__cons(n__0, y1)) → AND(isNatKind(0), n__isNatIListKind(activate(y1)))

The TRS R consists of the following rules:

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

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

The following pairs can be oriented strictly and are deleted.

ISNATILISTKIND(n__cons(n__nil, y1)) → AND(isNatKind(nil), n__isNatIListKind(activate(y1)))

The remaining pairs can at least be oriented weakly.

ISNATKIND(n__s(n__isNat(x0))) → ISNATKIND(isNat(x0))
ACTIVATE(n__and(n__isNatKind(x0), y1)) → AND(isNatKind(x0), y1)
ISNATKIND(n__s(x0)) → ISNATKIND(x0)
ISNATKIND(n__s(n__cons(x0, x1))) → ISNATKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(n__isNatKind(x0), y1)) → AND(isNatKind(isNatKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__length(x0), y1)) → AND(length(activate(x0)), y1)
ACTIVATE(n__and(n__and(x0, x1), y1)) → AND(and(activate(x0), x1), y1)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ISNATILISTKIND(n__cons(x0, y1)) → AND(isNatKind(x0), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → AND(isNatKind(length(activate(x0))), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__isNat(x0), y1)) → AND(isNat(x0), y1)
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATKIND(n__s(n__isNatKind(x0))) → ISNATKIND(isNatKind(x0))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → AND(isNatKind(s(activate(x0))), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(n__and(x0, x1))) → ISNATKIND(and(activate(x0), x1))
ISNATKIND(n__s(n__isNatIListKind(x0))) → ISNATKIND(isNatIListKind(x0))
ISNATILISTKIND(n__cons(n__zeros, y1)) → AND(isNatKind(zeros), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__cons(x0, x1), y1)) → AND(cons(activate(x0), x1), y1)
ISNATKIND(n__s(n__length(x0))) → ISNATKIND(length(activate(x0)))
ISNATKIND(n__s(n__s(x0))) → ISNATKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__and(x0, x1), y1)) → AND(isNatKind(and(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → AND(isNatKind(cons(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__isNatIListKind(x0), y1)) → AND(isNatKind(isNatIListKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__isNatIListKind(x0), y1)) → AND(isNatIListKind(x0), y1)
ACTIVATE(n__and(n__s(x0), y1)) → AND(s(activate(x0)), y1)
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(x0, y1)) → AND(x0, y1)
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ISNATILISTKIND(n__cons(n__isNat(x0), y1)) → AND(isNatKind(isNat(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATILISTKIND(n__cons(n__0, y1)) → AND(isNatKind(0), n__isNatIListKind(activate(y1)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(ACTIVATE(x₁)) = x₁
POL(AND(x₁, x₂)) = x₂
POL(ISNATILISTKIND(x₁)) = x₁
POL(ISNATKIND(x₁)) = x₁
POL(U11(x₁, x₂)) = 0
POL(U12(x₁)) = 0
POL(U21(x₁, x₂)) = 0
POL(U22(x₁)) = 0
POL(U51(x₁, x₂, x₃)) = 0
POL(U52(x₁, x₂)) = 0
POL(U53(x₁)) = 0
POL(U61(x₁, x₂)) = x₂
POL(activate(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = 0
POL(isNatIListKind(x₁)) = x₁
POL(isNatKind(x₁)) = x₁
POL(isNatList(x₁)) = 1 + x₁
POL(length(x₁)) = x₁
POL(n__0) = 0
POL(n__and(x₁, x₂)) = x₁ + x₂
POL(n__cons(x₁, x₂)) = x₁ + x₂
POL(n__isNat(x₁)) = 0
POL(n__isNatIListKind(x₁)) = x₁
POL(n__isNatKind(x₁)) = x₁
POL(n__length(x₁)) = x₁
POL(n__nil) = 1
POL(n__s(x₁)) = x₁
POL(n__zeros) = 0
POL(nil) = 1
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ 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

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ QDPOrderProof

                              ↳ QDP

                    ↳ QDP

          ↳ QDP

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

ISNATKIND(n__s(n__isNat(x0))) → ISNATKIND(isNat(x0))
ACTIVATE(n__and(n__isNatKind(x0), y1)) → AND(isNatKind(x0), y1)
ISNATKIND(n__s(x0)) → ISNATKIND(x0)
ISNATKIND(n__s(n__cons(x0, x1))) → ISNATKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(n__isNatKind(x0), y1)) → AND(isNatKind(isNatKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__length(x0), y1)) → AND(length(activate(x0)), y1)
ACTIVATE(n__and(n__and(x0, x1), y1)) → AND(and(activate(x0), x1), y1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ISNATILISTKIND(n__cons(x0, y1)) → AND(isNatKind(x0), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__isNat(x0), y1)) → AND(isNat(x0), y1)
ISNATILISTKIND(n__cons(n__length(x0), y1)) → AND(isNatKind(length(activate(x0))), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATKIND(n__s(n__isNatKind(x0))) → ISNATKIND(isNatKind(x0))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → AND(isNatKind(s(activate(x0))), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(n__and(x0, x1))) → ISNATKIND(and(activate(x0), x1))
ISNATKIND(n__s(n__isNatIListKind(x0))) → ISNATKIND(isNatIListKind(x0))
ISNATILISTKIND(n__cons(n__zeros, y1)) → AND(isNatKind(zeros), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__cons(x0, x1), y1)) → AND(cons(activate(x0), x1), y1)
ISNATKIND(n__s(n__length(x0))) → ISNATKIND(length(activate(x0)))
ISNATKIND(n__s(n__s(x0))) → ISNATKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__and(x0, x1), y1)) → AND(isNatKind(and(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → AND(isNatKind(cons(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__isNatIListKind(x0), y1)) → AND(isNatKind(isNatIListKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__s(x0), y1)) → AND(s(activate(x0)), y1)
ACTIVATE(n__and(n__isNatIListKind(x0), y1)) → AND(isNatIListKind(x0), y1)
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(x0, y1)) → AND(x0, y1)
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ISNATILISTKIND(n__cons(n__isNat(x0), y1)) → AND(isNatKind(isNat(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATILISTKIND(n__cons(n__0, y1)) → AND(isNatKind(0), n__isNatIListKind(activate(y1)))

The TRS R consists of the following rules:

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

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

The following pairs can be oriented strictly and are deleted.

ISNATILISTKIND(n__cons(n__zeros, y1)) → AND(isNatKind(zeros), n__isNatIListKind(activate(y1)))

The remaining pairs can at least be oriented weakly.

ISNATKIND(n__s(n__isNat(x0))) → ISNATKIND(isNat(x0))
ACTIVATE(n__and(n__isNatKind(x0), y1)) → AND(isNatKind(x0), y1)
ISNATKIND(n__s(x0)) → ISNATKIND(x0)
ISNATKIND(n__s(n__cons(x0, x1))) → ISNATKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(n__isNatKind(x0), y1)) → AND(isNatKind(isNatKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__length(x0), y1)) → AND(length(activate(x0)), y1)
ACTIVATE(n__and(n__and(x0, x1), y1)) → AND(and(activate(x0), x1), y1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ISNATILISTKIND(n__cons(x0, y1)) → AND(isNatKind(x0), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__isNat(x0), y1)) → AND(isNat(x0), y1)
ISNATILISTKIND(n__cons(n__length(x0), y1)) → AND(isNatKind(length(activate(x0))), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATKIND(n__s(n__isNatKind(x0))) → ISNATKIND(isNatKind(x0))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → AND(isNatKind(s(activate(x0))), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(n__and(x0, x1))) → ISNATKIND(and(activate(x0), x1))
ISNATKIND(n__s(n__isNatIListKind(x0))) → ISNATKIND(isNatIListKind(x0))
ACTIVATE(n__and(n__cons(x0, x1), y1)) → AND(cons(activate(x0), x1), y1)
ISNATKIND(n__s(n__length(x0))) → ISNATKIND(length(activate(x0)))
ISNATKIND(n__s(n__s(x0))) → ISNATKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__and(x0, x1), y1)) → AND(isNatKind(and(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → AND(isNatKind(cons(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__isNatIListKind(x0), y1)) → AND(isNatKind(isNatIListKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__s(x0), y1)) → AND(s(activate(x0)), y1)
ACTIVATE(n__and(n__isNatIListKind(x0), y1)) → AND(isNatIListKind(x0), y1)
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(x0, y1)) → AND(x0, y1)
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ISNATILISTKIND(n__cons(n__isNat(x0), y1)) → AND(isNatKind(isNat(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATILISTKIND(n__cons(n__0, y1)) → AND(isNatKind(0), n__isNatIListKind(activate(y1)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(ACTIVATE(x₁)) = x₁
POL(AND(x₁, x₂)) = x₂
POL(ISNATILISTKIND(x₁)) = x₁
POL(ISNATKIND(x₁)) = x₁
POL(U11(x₁, x₂)) = 0
POL(U12(x₁)) = 0
POL(U21(x₁, x₂)) = x₁
POL(U22(x₁)) = 0
POL(U51(x₁, x₂, x₃)) = x₃
POL(U52(x₁, x₂)) = x₂
POL(U53(x₁)) = x₁
POL(U61(x₁, x₂)) = x₂
POL(activate(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = x₁
POL(isNatIListKind(x₁)) = x₁
POL(isNatKind(x₁)) = x₁
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = x₁
POL(n__0) = 0
POL(n__and(x₁, x₂)) = x₁ + x₂
POL(n__cons(x₁, x₂)) = x₁ + x₂
POL(n__isNat(x₁)) = x₁
POL(n__isNatIListKind(x₁)) = x₁
POL(n__isNatKind(x₁)) = x₁
POL(n__length(x₁)) = x₁
POL(n__nil) = 0
POL(n__s(x₁)) = x₁
POL(n__zeros) = 1
POL(nil) = 0
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 1

The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ 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

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ QDPOrderProof

                              ↳ QDP

                    ↳ QDP

          ↳ QDP

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

ISNATKIND(n__s(n__isNat(x0))) → ISNATKIND(isNat(x0))
ACTIVATE(n__and(n__isNatKind(x0), y1)) → AND(isNatKind(x0), y1)
ISNATKIND(n__s(x0)) → ISNATKIND(x0)
ISNATKIND(n__s(n__cons(x0, x1))) → ISNATKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(n__isNatKind(x0), y1)) → AND(isNatKind(isNatKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__length(x0), y1)) → AND(length(activate(x0)), y1)
ACTIVATE(n__and(n__and(x0, x1), y1)) → AND(and(activate(x0), x1), y1)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ISNATILISTKIND(n__cons(x0, y1)) → AND(isNatKind(x0), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → AND(isNatKind(length(activate(x0))), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__isNat(x0), y1)) → AND(isNat(x0), y1)
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATKIND(n__s(n__isNatKind(x0))) → ISNATKIND(isNatKind(x0))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → AND(isNatKind(s(activate(x0))), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(n__and(x0, x1))) → ISNATKIND(and(activate(x0), x1))
ISNATKIND(n__s(n__isNatIListKind(x0))) → ISNATKIND(isNatIListKind(x0))
ACTIVATE(n__and(n__cons(x0, x1), y1)) → AND(cons(activate(x0), x1), y1)
ISNATKIND(n__s(n__length(x0))) → ISNATKIND(length(activate(x0)))
ISNATKIND(n__s(n__s(x0))) → ISNATKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__and(x0, x1), y1)) → AND(isNatKind(and(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → AND(isNatKind(cons(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__isNatIListKind(x0), y1)) → AND(isNatKind(isNatIListKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__s(x0), y1)) → AND(s(activate(x0)), y1)
ACTIVATE(n__and(n__isNatIListKind(x0), y1)) → AND(isNatIListKind(x0), y1)
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(x0, y1)) → AND(x0, y1)
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ISNATILISTKIND(n__cons(n__isNat(x0), y1)) → AND(isNatKind(isNat(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATILISTKIND(n__cons(n__0, y1)) → AND(isNatKind(0), n__isNatIListKind(activate(y1)))

The TRS R consists of the following rules:

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

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

The following pairs can be oriented strictly and are deleted.

ACTIVATE(n__and(n__isNat(x0), y1)) → AND(isNat(x0), y1)
ISNATILISTKIND(n__cons(n__isNat(x0), y1)) → AND(isNatKind(isNat(x0)), n__isNatIListKind(activate(y1)))

The remaining pairs can at least be oriented weakly.

ISNATKIND(n__s(n__isNat(x0))) → ISNATKIND(isNat(x0))
ACTIVATE(n__and(n__isNatKind(x0), y1)) → AND(isNatKind(x0), y1)
ISNATKIND(n__s(x0)) → ISNATKIND(x0)
ISNATKIND(n__s(n__cons(x0, x1))) → ISNATKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(n__isNatKind(x0), y1)) → AND(isNatKind(isNatKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__length(x0), y1)) → AND(length(activate(x0)), y1)
ACTIVATE(n__and(n__and(x0, x1), y1)) → AND(and(activate(x0), x1), y1)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ISNATILISTKIND(n__cons(x0, y1)) → AND(isNatKind(x0), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → AND(isNatKind(length(activate(x0))), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATKIND(n__s(n__isNatKind(x0))) → ISNATKIND(isNatKind(x0))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → AND(isNatKind(s(activate(x0))), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(n__and(x0, x1))) → ISNATKIND(and(activate(x0), x1))
ISNATKIND(n__s(n__isNatIListKind(x0))) → ISNATKIND(isNatIListKind(x0))
ACTIVATE(n__and(n__cons(x0, x1), y1)) → AND(cons(activate(x0), x1), y1)
ISNATKIND(n__s(n__length(x0))) → ISNATKIND(length(activate(x0)))
ISNATKIND(n__s(n__s(x0))) → ISNATKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__and(x0, x1), y1)) → AND(isNatKind(and(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → AND(isNatKind(cons(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__isNatIListKind(x0), y1)) → AND(isNatKind(isNatIListKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__s(x0), y1)) → AND(s(activate(x0)), y1)
ACTIVATE(n__and(n__isNatIListKind(x0), y1)) → AND(isNatIListKind(x0), y1)
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(x0, y1)) → AND(x0, y1)
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATILISTKIND(n__cons(n__0, y1)) → AND(isNatKind(0), n__isNatIListKind(activate(y1)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(ACTIVATE(x₁)) = x₁
POL(AND(x₁, x₂)) = x₂
POL(ISNATILISTKIND(x₁)) = x₁
POL(ISNATKIND(x₁)) = x₁
POL(U11(x₁, x₂)) = 0
POL(U12(x₁)) = 0
POL(U21(x₁, x₂)) = 1 + x₂
POL(U22(x₁)) = x₁
POL(U51(x₁, x₂, x₃)) = x₃
POL(U52(x₁, x₂)) = 0
POL(U53(x₁)) = 0
POL(U61(x₁, x₂)) = x₂
POL(activate(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = 1 + x₁
POL(isNatIListKind(x₁)) = x₁
POL(isNatKind(x₁)) = x₁
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = x₁
POL(n__0) = 0
POL(n__and(x₁, x₂)) = x₁ + x₂
POL(n__cons(x₁, x₂)) = x₁ + x₂
POL(n__isNat(x₁)) = 1 + x₁
POL(n__isNatIListKind(x₁)) = x₁
POL(n__isNatKind(x₁)) = x₁
POL(n__length(x₁)) = 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__isNat(X)) → isNat(X)
activate(X) → X
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U52(tt, V2) → U53(isNatList(activate(V2)))
U53(tt) → tt
U61(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
isNatKind(n__0) → tt
zeros → n__zeros
length(cons(N, L)) → U61(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))
length(nil) → 0
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNatIListKind(n__zeros) → tt
isNatIListKind(n__nil) → tt
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
and(tt, X) → activate(X)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__nil) → nil
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
0 → n__0
length(X) → n__length(X)
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ 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

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ QDPOrderProof

                              ↳ QDP

                    ↳ QDP

          ↳ QDP

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

ISNATKIND(n__s(n__isNat(x0))) → ISNATKIND(isNat(x0))
ACTIVATE(n__and(n__isNatKind(x0), y1)) → AND(isNatKind(x0), y1)
ISNATKIND(n__s(x0)) → ISNATKIND(x0)
ISNATKIND(n__s(n__cons(x0, x1))) → ISNATKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(n__isNatKind(x0), y1)) → AND(isNatKind(isNatKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__length(x0), y1)) → AND(length(activate(x0)), y1)
ACTIVATE(n__and(n__and(x0, x1), y1)) → AND(and(activate(x0), x1), y1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ISNATILISTKIND(n__cons(x0, y1)) → AND(isNatKind(x0), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__length(x0), y1)) → AND(isNatKind(length(activate(x0))), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATKIND(n__s(n__isNatKind(x0))) → ISNATKIND(isNatKind(x0))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → AND(isNatKind(s(activate(x0))), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(n__and(x0, x1))) → ISNATKIND(and(activate(x0), x1))
ISNATKIND(n__s(n__isNatIListKind(x0))) → ISNATKIND(isNatIListKind(x0))
ACTIVATE(n__and(n__cons(x0, x1), y1)) → AND(cons(activate(x0), x1), y1)
ISNATKIND(n__s(n__length(x0))) → ISNATKIND(length(activate(x0)))
ISNATKIND(n__s(n__s(x0))) → ISNATKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__and(x0, x1), y1)) → AND(isNatKind(and(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → AND(isNatKind(cons(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__isNatIListKind(x0), y1)) → AND(isNatKind(isNatIListKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__s(x0), y1)) → AND(s(activate(x0)), y1)
ACTIVATE(n__and(n__isNatIListKind(x0), y1)) → AND(isNatIListKind(x0), y1)
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(x0, y1)) → AND(x0, y1)
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATILISTKIND(n__cons(n__0, y1)) → AND(isNatKind(0), n__isNatIListKind(activate(y1)))

The TRS R consists of the following rules:

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

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

The following pairs can be oriented strictly and are deleted.

ACTIVATE(n__and(n__length(x0), y1)) → AND(length(activate(x0)), y1)
ISNATILISTKIND(n__cons(n__length(x0), y1)) → AND(isNatKind(length(activate(x0))), n__isNatIListKind(activate(y1)))

The remaining pairs can at least be oriented weakly.

ISNATKIND(n__s(n__isNat(x0))) → ISNATKIND(isNat(x0))
ACTIVATE(n__and(n__isNatKind(x0), y1)) → AND(isNatKind(x0), y1)
ISNATKIND(n__s(x0)) → ISNATKIND(x0)
ISNATKIND(n__s(n__cons(x0, x1))) → ISNATKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(n__isNatKind(x0), y1)) → AND(isNatKind(isNatKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__and(x0, x1), y1)) → AND(and(activate(x0), x1), y1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ISNATILISTKIND(n__cons(x0, y1)) → AND(isNatKind(x0), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATKIND(n__s(n__isNatKind(x0))) → ISNATKIND(isNatKind(x0))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → AND(isNatKind(s(activate(x0))), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(n__and(x0, x1))) → ISNATKIND(and(activate(x0), x1))
ISNATKIND(n__s(n__isNatIListKind(x0))) → ISNATKIND(isNatIListKind(x0))
ACTIVATE(n__and(n__cons(x0, x1), y1)) → AND(cons(activate(x0), x1), y1)
ISNATKIND(n__s(n__length(x0))) → ISNATKIND(length(activate(x0)))
ISNATKIND(n__s(n__s(x0))) → ISNATKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__and(x0, x1), y1)) → AND(isNatKind(and(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → AND(isNatKind(cons(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__isNatIListKind(x0), y1)) → AND(isNatKind(isNatIListKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__s(x0), y1)) → AND(s(activate(x0)), y1)
ACTIVATE(n__and(n__isNatIListKind(x0), y1)) → AND(isNatIListKind(x0), y1)
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(x0, y1)) → AND(x0, y1)
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATILISTKIND(n__cons(n__0, y1)) → AND(isNatKind(0), n__isNatIListKind(activate(y1)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(ACTIVATE(x₁)) = x₁
POL(AND(x₁, x₂)) = x₂
POL(ISNATILISTKIND(x₁)) = x₁
POL(ISNATKIND(x₁)) = x₁
POL(U11(x₁, x₂)) = 0
POL(U12(x₁)) = 0
POL(U21(x₁, x₂)) = 0
POL(U22(x₁)) = 0
POL(U51(x₁, x₂, x₃)) = 0
POL(U52(x₁, x₂)) = 0
POL(U53(x₁)) = 0
POL(U61(x₁, x₂)) = 1 + x₂
POL(activate(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = 0
POL(isNatIListKind(x₁)) = x₁
POL(isNatKind(x₁)) = x₁
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = 1 + x₁
POL(n__0) = 0
POL(n__and(x₁, x₂)) = x₁ + x₂
POL(n__cons(x₁, x₂)) = x₁ + x₂
POL(n__isNat(x₁)) = 0
POL(n__isNatIListKind(x₁)) = x₁
POL(n__isNatKind(x₁)) = x₁
POL(n__length(x₁)) = 1 + x₁
POL(n__nil) = 0
POL(n__s(x₁)) = x₁
POL(n__zeros) = 0
POL(nil) = 0
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ 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

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QDPOrderProof

                              ↳ QDP

                    ↳ QDP

          ↳ QDP

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

ISNATKIND(n__s(n__isNat(x0))) → ISNATKIND(isNat(x0))
ACTIVATE(n__and(n__isNatKind(x0), y1)) → AND(isNatKind(x0), y1)
ISNATKIND(n__s(x0)) → ISNATKIND(x0)
ISNATKIND(n__s(n__cons(x0, x1))) → ISNATKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(n__isNatKind(x0), y1)) → AND(isNatKind(isNatKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__and(x0, x1), y1)) → AND(and(activate(x0), x1), y1)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ISNATILISTKIND(n__cons(x0, y1)) → AND(isNatKind(x0), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATKIND(n__s(n__isNatKind(x0))) → ISNATKIND(isNatKind(x0))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → AND(isNatKind(s(activate(x0))), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(n__and(x0, x1))) → ISNATKIND(and(activate(x0), x1))
ISNATKIND(n__s(n__isNatIListKind(x0))) → ISNATKIND(isNatIListKind(x0))
ACTIVATE(n__and(n__cons(x0, x1), y1)) → AND(cons(activate(x0), x1), y1)
ISNATKIND(n__s(n__length(x0))) → ISNATKIND(length(activate(x0)))
ISNATKIND(n__s(n__s(x0))) → ISNATKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__and(x0, x1), y1)) → AND(isNatKind(and(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → AND(isNatKind(cons(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__isNatIListKind(x0), y1)) → AND(isNatKind(isNatIListKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__s(x0), y1)) → AND(s(activate(x0)), y1)
ACTIVATE(n__and(n__isNatIListKind(x0), y1)) → AND(isNatIListKind(x0), y1)
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(x0, y1)) → AND(x0, y1)
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATILISTKIND(n__cons(n__0, y1)) → AND(isNatKind(0), n__isNatIListKind(activate(y1)))

The TRS R consists of the following rules:

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

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

The following pairs can be oriented strictly and are deleted.

ISNATILISTKIND(n__cons(n__isNatIListKind(x0), y1)) → AND(isNatKind(isNatIListKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__isNatIListKind(x0), y1)) → AND(isNatIListKind(x0), y1)

The remaining pairs can at least be oriented weakly.

ISNATKIND(n__s(n__isNat(x0))) → ISNATKIND(isNat(x0))
ACTIVATE(n__and(n__isNatKind(x0), y1)) → AND(isNatKind(x0), y1)
ISNATKIND(n__s(x0)) → ISNATKIND(x0)
ISNATKIND(n__s(n__cons(x0, x1))) → ISNATKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(n__isNatKind(x0), y1)) → AND(isNatKind(isNatKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__and(x0, x1), y1)) → AND(and(activate(x0), x1), y1)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ISNATILISTKIND(n__cons(x0, y1)) → AND(isNatKind(x0), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATKIND(n__s(n__isNatKind(x0))) → ISNATKIND(isNatKind(x0))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → AND(isNatKind(s(activate(x0))), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(n__and(x0, x1))) → ISNATKIND(and(activate(x0), x1))
ISNATKIND(n__s(n__isNatIListKind(x0))) → ISNATKIND(isNatIListKind(x0))
ACTIVATE(n__and(n__cons(x0, x1), y1)) → AND(cons(activate(x0), x1), y1)
ISNATKIND(n__s(n__length(x0))) → ISNATKIND(length(activate(x0)))
ISNATKIND(n__s(n__s(x0))) → ISNATKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__and(x0, x1), y1)) → AND(isNatKind(and(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → AND(isNatKind(cons(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__s(x0), y1)) → AND(s(activate(x0)), y1)
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(x0, y1)) → AND(x0, y1)
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATILISTKIND(n__cons(n__0, y1)) → AND(isNatKind(0), n__isNatIListKind(activate(y1)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(ACTIVATE(x₁)) = x₁
POL(AND(x₁, x₂)) = x₂
POL(ISNATILISTKIND(x₁)) = 1 + x₁
POL(ISNATKIND(x₁)) = x₁
POL(U11(x₁, x₂)) = 0
POL(U12(x₁)) = 0
POL(U21(x₁, x₂)) = 0
POL(U22(x₁)) = 0
POL(U51(x₁, x₂, x₃)) = 0
POL(U52(x₁, x₂)) = 0
POL(U53(x₁)) = 0
POL(U61(x₁, x₂)) = 1 + x₂
POL(activate(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = 0
POL(isNatIListKind(x₁)) = 1 + x₁
POL(isNatKind(x₁)) = x₁
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 1 + x₁
POL(n__0) = 0
POL(n__and(x₁, x₂)) = x₁ + x₂
POL(n__cons(x₁, x₂)) = x₁ + x₂
POL(n__isNat(x₁)) = 0
POL(n__isNatIListKind(x₁)) = 1 + x₁
POL(n__isNatKind(x₁)) = x₁
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__isNat(X)) → isNat(X)
activate(X) → X
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U52(tt, V2) → U53(isNatList(activate(V2)))
U53(tt) → tt
U61(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
isNatKind(n__0) → tt
zeros → n__zeros
length(cons(N, L)) → U61(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(n__isNat(N), n__isNatKind(N))), activate(L))
length(nil) → 0
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNatIListKind(n__zeros) → tt
isNatIListKind(n__nil) → tt
activate(n__0) → 0
activate(n__length(X)) → length(activate(X))
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__and(X1, X2)) → and(activate(X1), X2)
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
and(tt, X) → activate(X)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__nil) → nil
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
0 → n__0
length(X) → n__length(X)
and(X1, X2) → n__and(X1, X2)
isNat(X) → n__isNat(X)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ 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

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QDPOrderProof

                              ↳ QDP

                    ↳ QDP

          ↳ QDP

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

ISNATKIND(n__s(n__isNat(x0))) → ISNATKIND(isNat(x0))
ACTIVATE(n__and(n__isNatKind(x0), y1)) → AND(isNatKind(x0), y1)
ISNATKIND(n__s(x0)) → ISNATKIND(x0)
ISNATKIND(n__s(n__cons(x0, x1))) → ISNATKIND(cons(activate(x0), x1))
ISNATILISTKIND(n__cons(n__isNatKind(x0), y1)) → AND(isNatKind(isNatKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__and(x0, x1), y1)) → AND(and(activate(x0), x1), y1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ISNATILISTKIND(n__cons(x0, y1)) → AND(isNatKind(x0), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATKIND(n__s(n__isNatKind(x0))) → ISNATKIND(isNatKind(x0))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → AND(isNatKind(s(activate(x0))), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(n__and(x0, x1))) → ISNATKIND(and(activate(x0), x1))
ISNATKIND(n__s(n__isNatIListKind(x0))) → ISNATKIND(isNatIListKind(x0))
ACTIVATE(n__and(n__cons(x0, x1), y1)) → AND(cons(activate(x0), x1), y1)
ISNATKIND(n__s(n__length(x0))) → ISNATKIND(length(activate(x0)))
ISNATKIND(n__s(n__s(x0))) → ISNATKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__and(x0, x1), y1)) → AND(isNatKind(and(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → AND(isNatKind(cons(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__s(x0), y1)) → AND(s(activate(x0)), y1)
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(x0, y1)) → AND(x0, y1)
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATILISTKIND(n__cons(n__0, y1)) → AND(isNatKind(0), n__isNatIListKind(activate(y1)))

The TRS R consists of the following rules:

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

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

The following pairs can be oriented strictly and are deleted.

ISNATKIND(n__s(n__cons(x0, x1))) → ISNATKIND(cons(activate(x0), x1))
ACTIVATE(n__and(n__cons(x0, x1), y1)) → AND(cons(activate(x0), x1), y1)

The remaining pairs can at least be oriented weakly.

ISNATKIND(n__s(n__isNat(x0))) → ISNATKIND(isNat(x0))
ACTIVATE(n__and(n__isNatKind(x0), y1)) → AND(isNatKind(x0), y1)
ISNATKIND(n__s(x0)) → ISNATKIND(x0)
ISNATILISTKIND(n__cons(n__isNatKind(x0), y1)) → AND(isNatKind(isNatKind(x0)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__and(x0, x1), y1)) → AND(and(activate(x0), x1), y1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ISNATILISTKIND(n__cons(x0, y1)) → AND(isNatKind(x0), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATKIND(n__s(n__isNatKind(x0))) → ISNATKIND(isNatKind(x0))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → AND(isNatKind(s(activate(x0))), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(n__and(x0, x1))) → ISNATKIND(and(activate(x0), x1))
ISNATKIND(n__s(n__isNatIListKind(x0))) → ISNATKIND(isNatIListKind(x0))
ISNATKIND(n__s(n__length(x0))) → ISNATKIND(length(activate(x0)))
ISNATKIND(n__s(n__s(x0))) → ISNATKIND(s(activate(x0)))
ISNATILISTKIND(n__cons(n__and(x0, x1), y1)) → AND(isNatKind(and(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → AND(isNatKind(cons(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__s(x0), y1)) → AND(s(activate(x0)), y1)
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(x0, y1)) → AND(x0, y1)
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATILISTKIND(n__cons(n__0, y1)) → AND(isNatKind(0), n__isNatIListKind(activate(y1)))

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

M( U52(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( n__isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( activate(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( and(x₁, x₂) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

/	1	0	\
\	0	1	/

x₂

M( n__nil ) =

/	0	\
\	1	/

M( n__s(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	1	1	/

x₁

M( U21(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( isNatIListKind(x₁) ) =

/	0	\
\	0	/

/	0	1	\
\	0	0	/

x₁

M( tt ) =

/	0	\
\	0	/

M( isNatList(x₁) ) =

/	0	\
\	1	/

/	0	0	\
\	0	1	/

x₁

M( zeros ) =

/	1	\
\	0	/

M( U12(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( s(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	1	1	/

x₁

M( isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U51(x₁, ..., x₃) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

/	0	0	\
\	0	1	/

x₃

M( nil ) =

/	0	\
\	1	/

M( n__length(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( n__isNatKind(x₁) ) =

/	0	\
\	0	/

/	0	1	\
\	0	0	/

x₁

M( n__zeros ) =

/	1	\
\	0	/

M( n__cons(x₁, x₂) ) =

/	1	\
\	0	/

/	0	0	\
\	0	1	/

x₁

/	0	0	\
\	0	1	/

x₂

M( U11(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( isNatKind(x₁) ) =

/	0	\
\	0	/

/	0	1	\
\	0	0	/

x₁

M( 0 ) =

/	0	\
\	0	/

M( n__and(x₁, x₂) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

/	1	0	\
\	0	1	/

x₂

M( cons(x₁, x₂) ) =

/	1	\
\	0	/

/	0	0	\
\	0	1	/

x₁

/	0	0	\
\	0	1	/

x₂

M( U22(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U53(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U61(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	1	/

x₂

M( n__0 ) =

/	0	\
\	0	/

M( n__isNatIListKind(x₁) ) =

/	0	\
\	0	/

/	0	1	\
\	0	0	/

x₁

M( length(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

Tuple symbols:

M( ISNATILISTKIND(x₁) ) =

[

]

x₁

M( ISNATKIND(x₁) ) =

[

]

x₁

M( AND(x₁, x₂) ) =

[

]

x₁

[

]

x₂

M( ACTIVATE(x₁) ) =

[

]

x₁

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ 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

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ QDPOrderProof

                              ↳ QDP

                    ↳ QDP

          ↳ QDP

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

ISNATKIND(n__s(n__isNat(x0))) → ISNATKIND(isNat(x0))
ACTIVATE(n__and(n__isNatKind(x0), y1)) → AND(isNatKind(x0), y1)
ISNATKIND(n__s(n__s(x0))) → ISNATKIND(s(activate(x0)))
ISNATKIND(n__s(n__length(x0))) → ISNATKIND(length(activate(x0)))
ISNATKIND(n__s(x0)) → ISNATKIND(x0)
ISNATILISTKIND(n__cons(n__and(x0, x1), y1)) → AND(isNatKind(and(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → AND(isNatKind(cons(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__s(x0), y1)) → AND(s(activate(x0)), y1)
ISNATILISTKIND(n__cons(n__isNatKind(x0), y1)) → AND(isNatKind(isNatKind(x0)), n__isNatIListKind(activate(y1)))
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(n__and(x0, x1), y1)) → AND(and(activate(x0), x1), y1)
ACTIVATE(n__and(x0, y1)) → AND(x0, y1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ISNATILISTKIND(n__cons(x0, y1)) → AND(isNatKind(x0), n__isNatIListKind(activate(y1)))
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATKIND(n__s(n__isNatKind(x0))) → ISNATKIND(isNatKind(x0))
ISNATILISTKIND(n__cons(n__0, y1)) → AND(isNatKind(0), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → AND(isNatKind(s(activate(x0))), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(n__and(x0, x1))) → ISNATKIND(and(activate(x0), x1))
ISNATKIND(n__s(n__isNatIListKind(x0))) → ISNATKIND(isNatIListKind(x0))

The TRS R consists of the following rules:

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

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

The following pairs can be oriented strictly and are deleted.

ISNATKIND(n__s(n__isNat(x0))) → ISNATKIND(isNat(x0))

The remaining pairs can at least be oriented weakly.

ACTIVATE(n__and(n__isNatKind(x0), y1)) → AND(isNatKind(x0), y1)
ISNATKIND(n__s(n__s(x0))) → ISNATKIND(s(activate(x0)))
ISNATKIND(n__s(n__length(x0))) → ISNATKIND(length(activate(x0)))
ISNATKIND(n__s(x0)) → ISNATKIND(x0)
ISNATILISTKIND(n__cons(n__and(x0, x1), y1)) → AND(isNatKind(and(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → AND(isNatKind(cons(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__s(x0), y1)) → AND(s(activate(x0)), y1)
ISNATILISTKIND(n__cons(n__isNatKind(x0), y1)) → AND(isNatKind(isNatKind(x0)), n__isNatIListKind(activate(y1)))
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(n__and(x0, x1), y1)) → AND(and(activate(x0), x1), y1)
ACTIVATE(n__and(x0, y1)) → AND(x0, y1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ISNATILISTKIND(n__cons(x0, y1)) → AND(isNatKind(x0), n__isNatIListKind(activate(y1)))
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATKIND(n__s(n__isNatKind(x0))) → ISNATKIND(isNatKind(x0))
ISNATILISTKIND(n__cons(n__0, y1)) → AND(isNatKind(0), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → AND(isNatKind(s(activate(x0))), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(n__and(x0, x1))) → ISNATKIND(and(activate(x0), x1))
ISNATKIND(n__s(n__isNatIListKind(x0))) → ISNATKIND(isNatIListKind(x0))

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

M( U52(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( n__isNat(x₁) ) =

/	1	\
\	1	/

/	0	0	\
\	1	1	/

x₁

M( activate(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( and(x₁, x₂) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

/	1	0	\
\	0	1	/

x₂

M( n__nil ) =

/	0	\
\	0	/

M( n__s(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	1	1	/

x₁

M( U21(x₁, x₂) ) =

/	1	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	1	0	/

x₂

M( isNatIListKind(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

M( tt ) =

/	0	\
\	0	/

M( isNatList(x₁) ) =

/	0	\
\	1	/

/	1	1	\
\	0	0	/

x₁

M( zeros ) =

/	0	\
\	0	/

M( U12(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( s(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	1	1	/

x₁

M( isNat(x₁) ) =

/	1	\
\	1	/

/	0	0	\
\	1	1	/

x₁

M( U51(x₁, ..., x₃) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

/	0	0	\
\	0	0	/

x₃

M( nil ) =

/	0	\
\	0	/

M( n__length(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	1	0	/

x₁

M( n__isNatKind(x₁) ) =

/	0	\
\	0	/

/	0	1	\
\	0	0	/

x₁

M( n__zeros ) =

/	0	\
\	0	/

M( n__cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	1	\
\	0	1	/

x₁

/	1	0	\
\	0	0	/

x₂

M( U11(x₁, x₂) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( isNatKind(x₁) ) =

/	0	\
\	0	/

/	0	1	\
\	0	0	/

x₁

M( 0 ) =

/	0	\
\	0	/

M( n__and(x₁, x₂) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

/	1	0	\
\	0	1	/

x₂

M( cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	1	\
\	0	1	/

x₁

/	1	0	\
\	0	0	/

x₂

M( U22(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U53(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U61(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	1	0	/

x₂

M( n__0 ) =

/	0	\
\	0	/

M( n__isNatIListKind(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

M( length(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	1	0	/

x₁

Tuple symbols:

M( ISNATILISTKIND(x₁) ) =

[

]

x₁

M( ISNATKIND(x₁) ) =

[

]

x₁

M( AND(x₁, x₂) ) =

[

]

x₁

[

]

x₂

M( ACTIVATE(x₁) ) =

[

]

x₁

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ 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

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ QDPOrderProof

                              ↳ QDP

                    ↳ QDP

          ↳ QDP

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

ACTIVATE(n__and(n__isNatKind(x0), y1)) → AND(isNatKind(x0), y1)
ISNATKIND(n__s(n__s(x0))) → ISNATKIND(s(activate(x0)))
ISNATKIND(n__s(n__length(x0))) → ISNATKIND(length(activate(x0)))
ISNATKIND(n__s(x0)) → ISNATKIND(x0)
ISNATILISTKIND(n__cons(n__and(x0, x1), y1)) → AND(isNatKind(and(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → AND(isNatKind(cons(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__s(x0), y1)) → AND(s(activate(x0)), y1)
ISNATILISTKIND(n__cons(n__isNatKind(x0), y1)) → AND(isNatKind(isNatKind(x0)), n__isNatIListKind(activate(y1)))
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(n__and(x0, x1), y1)) → AND(and(activate(x0), x1), y1)
ACTIVATE(n__and(x0, y1)) → AND(x0, y1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ISNATILISTKIND(n__cons(x0, y1)) → AND(isNatKind(x0), n__isNatIListKind(activate(y1)))
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATKIND(n__s(n__isNatKind(x0))) → ISNATKIND(isNatKind(x0))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → AND(isNatKind(s(activate(x0))), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__0, y1)) → AND(isNatKind(0), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(n__and(x0, x1))) → ISNATKIND(and(activate(x0), x1))
ISNATKIND(n__s(n__isNatIListKind(x0))) → ISNATKIND(isNatIListKind(x0))

The TRS R consists of the following rules:

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

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

The following pairs can be oriented strictly and are deleted.

ISNATILISTKIND(n__cons(n__cons(x0, x1), y1)) → AND(isNatKind(cons(activate(x0), x1)), n__isNatIListKind(activate(y1)))

The remaining pairs can at least be oriented weakly.

ACTIVATE(n__and(n__isNatKind(x0), y1)) → AND(isNatKind(x0), y1)
ISNATKIND(n__s(n__s(x0))) → ISNATKIND(s(activate(x0)))
ISNATKIND(n__s(n__length(x0))) → ISNATKIND(length(activate(x0)))
ISNATKIND(n__s(x0)) → ISNATKIND(x0)
ISNATILISTKIND(n__cons(n__and(x0, x1), y1)) → AND(isNatKind(and(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__s(x0), y1)) → AND(s(activate(x0)), y1)
ISNATILISTKIND(n__cons(n__isNatKind(x0), y1)) → AND(isNatKind(isNatKind(x0)), n__isNatIListKind(activate(y1)))
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(n__and(x0, x1), y1)) → AND(and(activate(x0), x1), y1)
ACTIVATE(n__and(x0, y1)) → AND(x0, y1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ISNATILISTKIND(n__cons(x0, y1)) → AND(isNatKind(x0), n__isNatIListKind(activate(y1)))
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATKIND(n__s(n__isNatKind(x0))) → ISNATKIND(isNatKind(x0))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → AND(isNatKind(s(activate(x0))), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__0, y1)) → AND(isNatKind(0), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(n__and(x0, x1))) → ISNATKIND(and(activate(x0), x1))
ISNATKIND(n__s(n__isNatIListKind(x0))) → ISNATKIND(isNatIListKind(x0))

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

M( U52(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( n__isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( activate(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

/	1	0	\
\	0	1	/

x₂

M( n__nil ) =

/	1	\
\	0	/

M( n__s(x₁) ) =

/	0	\
\	0	/

/	1	1	\
\	0	1	/

x₁

M( U21(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( isNatIListKind(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	1	0	/

x₁

M( tt ) =

/	0	\
\	0	/

M( isNatList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	1	0	/

x₁

M( zeros ) =

/	0	\
\	1	/

M( U12(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( s(x₁) ) =

/	0	\
\	0	/

/	1	1	\
\	0	1	/

x₁

M( isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U51(x₁, ..., x₃) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

/	0	0	\
\	0	0	/

x₃

M( nil ) =

/	1	\
\	0	/

M( n__length(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

M( n__isNatKind(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	1	0	/

x₁

M( n__zeros ) =

/	0	\
\	1	/

M( n__cons(x₁, x₂) ) =

/	0	\
\	1	/

/	1	1	\
\	0	0	/

x₁

/	1	0	\
\	0	0	/

x₂

M( U11(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( isNatKind(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	1	0	/

x₁

M( 0 ) =

/	0	\
\	0	/

M( n__and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

/	1	0	\
\	0	1	/

x₂

M( cons(x₁, x₂) ) =

/	0	\
\	1	/

/	1	1	\
\	0	0	/

x₁

/	1	0	\
\	0	0	/

x₂

M( U22(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U53(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U61(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	0	0	/

x₂

M( n__0 ) =

/	0	\
\	0	/

M( n__isNatIListKind(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	1	0	/

x₁

M( length(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

Tuple symbols:

M( ISNATILISTKIND(x₁) ) =

[

]

x₁

M( AND(x₁, x₂) ) =

[

]

x₁

[

]

x₂

M( ISNATKIND(x₁) ) =

[

]

x₁

M( ACTIVATE(x₁) ) =

[

]

x₁

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ 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

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ QDPOrderProof

                              ↳ QDP

                    ↳ QDP

          ↳ QDP

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

ACTIVATE(n__and(n__isNatKind(x0), y1)) → AND(isNatKind(x0), y1)
ISNATKIND(n__s(n__s(x0))) → ISNATKIND(s(activate(x0)))
ISNATKIND(n__s(n__length(x0))) → ISNATKIND(length(activate(x0)))
ISNATKIND(n__s(x0)) → ISNATKIND(x0)
ISNATILISTKIND(n__cons(n__and(x0, x1), y1)) → AND(isNatKind(and(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__s(x0), y1)) → AND(s(activate(x0)), y1)
ISNATILISTKIND(n__cons(n__isNatKind(x0), y1)) → AND(isNatKind(isNatKind(x0)), n__isNatIListKind(activate(y1)))
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(n__and(x0, x1), y1)) → AND(and(activate(x0), x1), y1)
ACTIVATE(n__and(x0, y1)) → AND(x0, y1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ISNATILISTKIND(n__cons(x0, y1)) → AND(isNatKind(x0), n__isNatIListKind(activate(y1)))
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATKIND(n__s(n__isNatKind(x0))) → ISNATKIND(isNatKind(x0))
ISNATILISTKIND(n__cons(n__0, y1)) → AND(isNatKind(0), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → AND(isNatKind(s(activate(x0))), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(n__and(x0, x1))) → ISNATKIND(and(activate(x0), x1))
ISNATKIND(n__s(n__isNatIListKind(x0))) → ISNATKIND(isNatIListKind(x0))

The TRS R consists of the following rules:

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

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

The following pairs can be oriented strictly and are deleted.

ISNATILISTKIND(n__cons(n__isNatKind(x0), y1)) → AND(isNatKind(isNatKind(x0)), n__isNatIListKind(activate(y1)))

The remaining pairs can at least be oriented weakly.

ACTIVATE(n__and(n__isNatKind(x0), y1)) → AND(isNatKind(x0), y1)
ISNATKIND(n__s(n__s(x0))) → ISNATKIND(s(activate(x0)))
ISNATKIND(n__s(n__length(x0))) → ISNATKIND(length(activate(x0)))
ISNATKIND(n__s(x0)) → ISNATKIND(x0)
ISNATILISTKIND(n__cons(n__and(x0, x1), y1)) → AND(isNatKind(and(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__s(x0), y1)) → AND(s(activate(x0)), y1)
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(n__and(x0, x1), y1)) → AND(and(activate(x0), x1), y1)
ACTIVATE(n__and(x0, y1)) → AND(x0, y1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ISNATILISTKIND(n__cons(x0, y1)) → AND(isNatKind(x0), n__isNatIListKind(activate(y1)))
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATKIND(n__s(n__isNatKind(x0))) → ISNATKIND(isNatKind(x0))
ISNATILISTKIND(n__cons(n__0, y1)) → AND(isNatKind(0), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → AND(isNatKind(s(activate(x0))), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(n__and(x0, x1))) → ISNATKIND(and(activate(x0), x1))
ISNATKIND(n__s(n__isNatIListKind(x0))) → ISNATKIND(isNatIListKind(x0))

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

M( U52(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( n__isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( activate(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

/	1	0	\
\	0	1	/

x₂

M( n__nil ) =

/	0	\
\	0	/

M( n__s(x₁) ) =

/	0	\
\	0	/

/	1	1	\
\	0	1	/

x₁

M( U21(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( isNatIListKind(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( tt ) =

/	0	\
\	0	/

M( isNatList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	1	0	/

x₁

M( zeros ) =

/	1	\
\	0	/

M( U12(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( s(x₁) ) =

/	0	\
\	0	/

/	1	1	\
\	0	1	/

x₁

M( isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U51(x₁, ..., x₃) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

/	0	0	\
\	0	0	/

x₃

M( nil ) =

/	0	\
\	0	/

M( n__length(x₁) ) =

/	0	\
\	0	/

/	0	1	\
\	0	0	/

x₁

M( n__isNatKind(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	1	0	/

x₁

M( n__zeros ) =

/	1	\
\	0	/

M( n__cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	1	0	/

x₁

/	0	0	\
\	0	1	/

x₂

M( U11(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( isNatKind(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	1	0	/

x₁

M( 0 ) =

/	0	\
\	0	/

M( n__and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

/	1	0	\
\	0	1	/

x₂

M( cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	1	0	/

x₁

/	0	0	\
\	0	1	/

x₂

M( U22(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U53(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U61(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	1	\
\	0	0	/

x₂

M( n__0 ) =

/	0	\
\	0	/

M( n__isNatIListKind(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( length(x₁) ) =

/	0	\
\	0	/

/	0	1	\
\	0	0	/

x₁

Tuple symbols:

M( ISNATILISTKIND(x₁) ) =

[

]

x₁

M( AND(x₁, x₂) ) =

[

]

x₁

[

]

x₂

M( ISNATKIND(x₁) ) =

[

]

x₁

M( ACTIVATE(x₁) ) =

[

]

x₁

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ 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

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ QDPOrderProof

                              ↳ QDP

                    ↳ QDP

          ↳ QDP

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

ACTIVATE(n__and(n__isNatKind(x0), y1)) → AND(isNatKind(x0), y1)
ISNATKIND(n__s(n__s(x0))) → ISNATKIND(s(activate(x0)))
ISNATKIND(n__s(n__length(x0))) → ISNATKIND(length(activate(x0)))
ISNATKIND(n__s(x0)) → ISNATKIND(x0)
ISNATILISTKIND(n__cons(n__and(x0, x1), y1)) → AND(isNatKind(and(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__s(x0), y1)) → AND(s(activate(x0)), y1)
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(n__and(x0, x1), y1)) → AND(and(activate(x0), x1), y1)
ACTIVATE(n__and(x0, y1)) → AND(x0, y1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ISNATILISTKIND(n__cons(x0, y1)) → AND(isNatKind(x0), n__isNatIListKind(activate(y1)))
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATKIND(n__s(n__isNatKind(x0))) → ISNATKIND(isNatKind(x0))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → AND(isNatKind(s(activate(x0))), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__0, y1)) → AND(isNatKind(0), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(n__and(x0, x1))) → ISNATKIND(and(activate(x0), x1))
ISNATKIND(n__s(n__isNatIListKind(x0))) → ISNATKIND(isNatIListKind(x0))

The TRS R consists of the following rules:

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

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

The following pairs can be oriented strictly and are deleted.

ISNATKIND(n__s(n__isNatIListKind(x0))) → ISNATKIND(isNatIListKind(x0))

The remaining pairs can at least be oriented weakly.

ACTIVATE(n__and(n__isNatKind(x0), y1)) → AND(isNatKind(x0), y1)
ISNATKIND(n__s(n__s(x0))) → ISNATKIND(s(activate(x0)))
ISNATKIND(n__s(n__length(x0))) → ISNATKIND(length(activate(x0)))
ISNATKIND(n__s(x0)) → ISNATKIND(x0)
ISNATILISTKIND(n__cons(n__and(x0, x1), y1)) → AND(isNatKind(and(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__s(x0), y1)) → AND(s(activate(x0)), y1)
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(n__and(x0, x1), y1)) → AND(and(activate(x0), x1), y1)
ACTIVATE(n__and(x0, y1)) → AND(x0, y1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ISNATILISTKIND(n__cons(x0, y1)) → AND(isNatKind(x0), n__isNatIListKind(activate(y1)))
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATKIND(n__s(n__isNatKind(x0))) → ISNATKIND(isNatKind(x0))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → AND(isNatKind(s(activate(x0))), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__0, y1)) → AND(isNatKind(0), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(n__and(x0, x1))) → ISNATKIND(and(activate(x0), x1))

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

M( U52(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( n__isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( activate(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( and(x₁, x₂) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

/	1	0	\
\	0	1	/

x₂

M( n__nil ) =

/	0	\
\	0	/

M( n__s(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	1	1	/

x₁

M( U21(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( isNatIListKind(x₁) ) =

/	1	\
\	0	/

/	0	1	\
\	0	0	/

x₁

M( tt ) =

/	0	\
\	0	/

M( isNatList(x₁) ) =

/	1	\
\	1	/

/	1	1	\
\	1	1	/

x₁

M( zeros ) =

/	0	\
\	0	/

M( U12(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( s(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	1	1	/

x₁

M( isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U51(x₁, ..., x₃) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

/	0	0	\
\	0	0	/

x₃

M( nil ) =

/	0	\
\	0	/

M( n__length(x₁) ) =

/	0	\
\	1	/

/	0	0	\
\	0	1	/

x₁

M( n__isNatKind(x₁) ) =

/	0	\
\	0	/

/	0	1	\
\	0	0	/

x₁

M( n__zeros ) =

/	0	\
\	0	/

M( n__cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

/	0	0	\
\	0	1	/

x₂

M( U11(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( isNatKind(x₁) ) =

/	0	\
\	0	/

/	0	1	\
\	0	0	/

x₁

M( 0 ) =

/	0	\
\	0	/

M( n__and(x₁, x₂) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

/	1	0	\
\	0	1	/

x₂

M( cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

/	0	0	\
\	0	1	/

x₂

M( U22(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U53(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U61(x₁, x₂) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	1	/

x₂

M( n__0 ) =

/	0	\
\	0	/

M( n__isNatIListKind(x₁) ) =

/	1	\
\	0	/

/	0	1	\
\	0	0	/

x₁

M( length(x₁) ) =

/	0	\
\	1	/

/	0	0	\
\	0	1	/

x₁

Tuple symbols:

M( ISNATILISTKIND(x₁) ) =

[

]

x₁

M( AND(x₁, x₂) ) =

[

]

x₁

[

]

x₂

M( ISNATKIND(x₁) ) =

[

]

x₁

M( ACTIVATE(x₁) ) =

[

]

x₁

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ 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

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ QDPOrderProof

                              ↳ QDP

                    ↳ QDP

          ↳ QDP

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

ACTIVATE(n__and(n__isNatKind(x0), y1)) → AND(isNatKind(x0), y1)
ISNATKIND(n__s(n__s(x0))) → ISNATKIND(s(activate(x0)))
ISNATKIND(n__s(n__length(x0))) → ISNATKIND(length(activate(x0)))
ISNATKIND(n__s(x0)) → ISNATKIND(x0)
ISNATILISTKIND(n__cons(n__and(x0, x1), y1)) → AND(isNatKind(and(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__s(x0), y1)) → AND(s(activate(x0)), y1)
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(n__and(x0, x1), y1)) → AND(and(activate(x0), x1), y1)
ACTIVATE(n__and(x0, y1)) → AND(x0, y1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ISNATILISTKIND(n__cons(x0, y1)) → AND(isNatKind(x0), n__isNatIListKind(activate(y1)))
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATKIND(n__s(n__isNatKind(x0))) → ISNATKIND(isNatKind(x0))
ISNATILISTKIND(n__cons(n__0, y1)) → AND(isNatKind(0), n__isNatIListKind(activate(y1)))
ISNATILISTKIND(n__cons(n__s(x0), y1)) → AND(isNatKind(s(activate(x0))), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(n__and(x0, x1))) → ISNATKIND(and(activate(x0), x1))

The TRS R consists of the following rules:

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

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

The following pairs can be oriented strictly and are deleted.

ISNATILISTKIND(n__cons(n__s(x0), y1)) → AND(isNatKind(s(activate(x0))), n__isNatIListKind(activate(y1)))

The remaining pairs can at least be oriented weakly.

ACTIVATE(n__and(n__isNatKind(x0), y1)) → AND(isNatKind(x0), y1)
ISNATKIND(n__s(n__s(x0))) → ISNATKIND(s(activate(x0)))
ISNATKIND(n__s(n__length(x0))) → ISNATKIND(length(activate(x0)))
ISNATKIND(n__s(x0)) → ISNATKIND(x0)
ISNATILISTKIND(n__cons(n__and(x0, x1), y1)) → AND(isNatKind(and(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__s(x0), y1)) → AND(s(activate(x0)), y1)
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(n__and(x0, x1), y1)) → AND(and(activate(x0), x1), y1)
ACTIVATE(n__and(x0, y1)) → AND(x0, y1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ISNATILISTKIND(n__cons(x0, y1)) → AND(isNatKind(x0), n__isNatIListKind(activate(y1)))
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATKIND(n__s(n__isNatKind(x0))) → ISNATKIND(isNatKind(x0))
ISNATILISTKIND(n__cons(n__0, y1)) → AND(isNatKind(0), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(n__and(x0, x1))) → ISNATKIND(and(activate(x0), x1))

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

M( U52(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( n__isNat(x₁) ) =

/	0	\
\	0	/

/	1	1	\
\	0	0	/

x₁

M( activate(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

/	1	0	\
\	0	1	/

x₂

M( n__nil ) =

/	0	\
\	0	/

M( n__s(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( U21(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( isNatIListKind(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	1	0	/

x₁

M( tt ) =

/	0	\
\	0	/

M( isNatList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( zeros ) =

/	0	\
\	0	/

M( U12(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( s(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( isNat(x₁) ) =

/	0	\
\	0	/

/	1	1	\
\	0	0	/

x₁

M( U51(x₁, ..., x₃) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

/	0	0	\
\	0	0	/

x₃

M( nil ) =

/	0	\
\	0	/

M( n__length(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	1	0	/

x₁

M( n__isNatKind(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( n__zeros ) =

/	0	\
\	0	/

M( n__cons(x₁, x₂) ) =

/	0	\
\	0	/

/	1	1	\
\	0	0	/

x₁

/	1	0	\
\	0	0	/

x₂

M( U11(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( isNatKind(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( 0 ) =

/	0	\
\	0	/

M( n__and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

/	1	0	\
\	0	1	/

x₂

M( cons(x₁, x₂) ) =

/	0	\
\	0	/

/	1	1	\
\	0	0	/

x₁

/	1	0	\
\	0	0	/

x₂

M( U22(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U53(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U61(x₁, x₂) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	1	0	/

x₂

M( n__0 ) =

/	0	\
\	0	/

M( n__isNatIListKind(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	1	0	/

x₁

M( length(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	1	0	/

x₁

Tuple symbols:

M( ISNATILISTKIND(x₁) ) =

[

]

x₁

M( AND(x₁, x₂) ) =

[

]

x₁

[

]

x₂

M( ISNATKIND(x₁) ) =

[

]

x₁

M( ACTIVATE(x₁) ) =

[

]

x₁

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ 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

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ QDPOrderProof

                              ↳ QDP

                    ↳ QDP

          ↳ QDP

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

ACTIVATE(n__and(n__isNatKind(x0), y1)) → AND(isNatKind(x0), y1)
ISNATKIND(n__s(n__s(x0))) → ISNATKIND(s(activate(x0)))
ISNATKIND(n__s(n__length(x0))) → ISNATKIND(length(activate(x0)))
ISNATKIND(n__s(x0)) → ISNATKIND(x0)
ISNATILISTKIND(n__cons(n__and(x0, x1), y1)) → AND(isNatKind(and(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__s(x0), y1)) → AND(s(activate(x0)), y1)
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(n__and(x0, x1), y1)) → AND(and(activate(x0), x1), y1)
ACTIVATE(n__and(x0, y1)) → AND(x0, y1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ISNATILISTKIND(n__cons(x0, y1)) → AND(isNatKind(x0), n__isNatIListKind(activate(y1)))
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATKIND(n__s(n__isNatKind(x0))) → ISNATKIND(isNatKind(x0))
ISNATILISTKIND(n__cons(n__0, y1)) → AND(isNatKind(0), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(n__and(x0, x1))) → ISNATKIND(and(activate(x0), x1))

The TRS R consists of the following rules:

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

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

The following pairs can be oriented strictly and are deleted.

ISNATKIND(n__s(n__isNatKind(x0))) → ISNATKIND(isNatKind(x0))

The remaining pairs can at least be oriented weakly.

ACTIVATE(n__and(n__isNatKind(x0), y1)) → AND(isNatKind(x0), y1)
ISNATKIND(n__s(n__s(x0))) → ISNATKIND(s(activate(x0)))
ISNATKIND(n__s(n__length(x0))) → ISNATKIND(length(activate(x0)))
ISNATKIND(n__s(x0)) → ISNATKIND(x0)
ISNATILISTKIND(n__cons(n__and(x0, x1), y1)) → AND(isNatKind(and(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__s(x0), y1)) → AND(s(activate(x0)), y1)
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(n__and(x0, x1), y1)) → AND(and(activate(x0), x1), y1)
ACTIVATE(n__and(x0, y1)) → AND(x0, y1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ISNATILISTKIND(n__cons(x0, y1)) → AND(isNatKind(x0), n__isNatIListKind(activate(y1)))
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATILISTKIND(n__cons(n__0, y1)) → AND(isNatKind(0), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(n__and(x0, x1))) → ISNATKIND(and(activate(x0), x1))

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

M( U52(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( n__isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( activate(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( and(x₁, x₂) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

/	1	0	\
\	0	1	/

x₂

M( n__nil ) =

/	0	\
\	0	/

M( n__s(x₁) ) =

/	0	\
\	0	/

/	1	1	\
\	0	0	/

x₁

M( U21(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( isNatIListKind(x₁) ) =

/	0	\
\	0	/

/	0	1	\
\	0	0	/

x₁

M( tt ) =

/	0	\
\	0	/

M( isNatList(x₁) ) =

/	0	\
\	1	/

/	0	1	\
\	0	0	/

x₁

M( zeros ) =

/	0	\
\	0	/

M( U12(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( s(x₁) ) =

/	0	\
\	0	/

/	1	1	\
\	0	0	/

x₁

M( isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U51(x₁, ..., x₃) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	1	\
\	0	0	/

x₂

/	0	0	\
\	0	0	/

x₃

M( nil ) =

/	0	\
\	0	/

M( n__length(x₁) ) =

/	0	\
\	0	/

/	0	1	\
\	0	0	/

x₁

M( n__isNatKind(x₁) ) =

/	0	\
\	1	/

/	1	0	\
\	0	0	/

x₁

M( n__zeros ) =

/	0	\
\	0	/

M( n__cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	1	1	/

x₁

/	0	0	\
\	0	1	/

x₂

M( U11(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( isNatKind(x₁) ) =

/	0	\
\	1	/

/	1	0	\
\	0	0	/

x₁

M( 0 ) =

/	0	\
\	0	/

M( n__and(x₁, x₂) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

/	1	0	\
\	0	1	/

x₂

M( cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	1	1	/

x₁

/	0	0	\
\	0	1	/

x₂

M( U22(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U53(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U61(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	1	\
\	0	0	/

x₂

M( n__0 ) =

/	0	\
\	0	/

M( n__isNatIListKind(x₁) ) =

/	0	\
\	0	/

/	0	1	\
\	0	0	/

x₁

M( length(x₁) ) =

/	0	\
\	0	/

/	0	1	\
\	0	0	/

x₁

Tuple symbols:

M( ISNATILISTKIND(x₁) ) =

[

]

x₁

M( AND(x₁, x₂) ) =

[

]

x₁

[

]

x₂

M( ISNATKIND(x₁) ) =

[

]

x₁

M( ACTIVATE(x₁) ) =

[

]

x₁

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ 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

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ QDPOrderProof

                              ↳ QDP

                    ↳ QDP

          ↳ QDP

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

ACTIVATE(n__and(n__isNatKind(x0), y1)) → AND(isNatKind(x0), y1)
ISNATKIND(n__s(n__s(x0))) → ISNATKIND(s(activate(x0)))
ISNATKIND(n__s(n__length(x0))) → ISNATKIND(length(activate(x0)))
ISNATKIND(n__s(x0)) → ISNATKIND(x0)
ISNATILISTKIND(n__cons(n__and(x0, x1), y1)) → AND(isNatKind(and(activate(x0), x1)), n__isNatIListKind(activate(y1)))
ACTIVATE(n__and(n__s(x0), y1)) → AND(s(activate(x0)), y1)
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(n__and(x0, x1), y1)) → AND(and(activate(x0), x1), y1)
ACTIVATE(n__and(x0, y1)) → AND(x0, y1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ISNATILISTKIND(n__cons(x0, y1)) → AND(isNatKind(x0), n__isNatIListKind(activate(y1)))
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATILISTKIND(n__cons(n__0, y1)) → AND(isNatKind(0), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(n__and(x0, x1))) → ISNATKIND(and(activate(x0), x1))

The TRS R consists of the following rules:

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

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

The following pairs can be oriented strictly and are deleted.

ACTIVATE(n__and(n__s(x0), y1)) → AND(s(activate(x0)), y1)

The remaining pairs can at least be oriented weakly.

ACTIVATE(n__and(n__isNatKind(x0), y1)) → AND(isNatKind(x0), y1)
ISNATKIND(n__s(n__s(x0))) → ISNATKIND(s(activate(x0)))
ISNATKIND(n__s(n__length(x0))) → ISNATKIND(length(activate(x0)))
ISNATKIND(n__s(x0)) → ISNATKIND(x0)
ISNATILISTKIND(n__cons(n__and(x0, x1), y1)) → AND(isNatKind(and(activate(x0), x1)), n__isNatIListKind(activate(y1)))
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(n__and(x0, x1), y1)) → AND(and(activate(x0), x1), y1)
ACTIVATE(n__and(x0, y1)) → AND(x0, y1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ISNATILISTKIND(n__cons(x0, y1)) → AND(isNatKind(x0), n__isNatIListKind(activate(y1)))
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATILISTKIND(n__cons(n__0, y1)) → AND(isNatKind(0), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(n__and(x0, x1))) → ISNATKIND(and(activate(x0), x1))

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

M( U52(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( n__isNat(x₁) ) =

/	0	\
\	0	/

/	0	1	\
\	0	1	/

x₁

M( activate(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	1	1	/

x₁

/	1	0	\
\	0	1	/

x₂

M( n__nil ) =

/	0	\
\	0	/

M( n__s(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( U21(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( isNatIListKind(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( tt ) =

/	0	\
\	0	/

M( isNatList(x₁) ) =

/	0	\
\	0	/

/	0	1	\
\	0	0	/

x₁

M( zeros ) =

/	0	\
\	0	/

M( U12(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( s(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( isNat(x₁) ) =

/	0	\
\	0	/

/	0	1	\
\	0	1	/

x₁

M( U51(x₁, ..., x₃) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

/	0	0	\
\	0	0	/

x₃

M( nil ) =

/	0	\
\	0	/

M( n__length(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( n__isNatKind(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( n__zeros ) =

/	0	\
\	0	/

M( n__cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

/	0	0	\
\	0	1	/

x₂

M( U11(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( isNatKind(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( 0 ) =

/	0	\
\	0	/

M( n__and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	1	1	/

x₁

/	1	0	\
\	0	1	/

x₂

M( cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

/	0	0	\
\	0	1	/

x₂

M( U22(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U53(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( U61(x₁, x₂) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	1	/

x₂

M( n__0 ) =

/	0	\
\	0	/

M( n__isNatIListKind(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( length(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	0	1	/

x₁

Tuple symbols:

M( ISNATILISTKIND(x₁) ) =

[

]

x₁

M( AND(x₁, x₂) ) =

[

]

x₁

[

]

x₂

M( ISNATKIND(x₁) ) =

[

]

x₁

M( ACTIVATE(x₁) ) =

[

]

x₁

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ 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

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ SemLabProof

                                                                                                                                                                                            ↳ SemLabProof2

                              ↳ QDP

                    ↳ QDP

          ↳ QDP

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

ACTIVATE(n__and(n__isNatKind(x0), y1)) → AND(isNatKind(x0), y1)
ISNATKIND(n__s(n__s(x0))) → ISNATKIND(s(activate(x0)))
ISNATKIND(n__s(n__length(x0))) → ISNATKIND(length(activate(x0)))
ISNATKIND(n__s(x0)) → ISNATKIND(x0)
ISNATILISTKIND(n__cons(n__and(x0, x1), y1)) → AND(isNatKind(and(activate(x0), x1)), n__isNatIListKind(activate(y1)))
AND(tt, X) → ACTIVATE(X)
ACTIVATE(n__and(n__and(x0, x1), y1)) → AND(and(activate(x0), x1), y1)
ACTIVATE(n__and(x0, y1)) → AND(x0, y1)
ACTIVATE(n__s(X)) → ACTIVATE(X)
ACTIVATE(n__and(X1, X2)) → ACTIVATE(X1)
ISNATILISTKIND(n__cons(x0, y1)) → AND(isNatKind(x0), n__isNatIListKind(activate(y1)))
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATILISTKIND(n__cons(n__0, y1)) → AND(isNatKind(0), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(n__and(x0, x1))) → ISNATKIND(and(activate(x0), x1))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We found the following model for the rules of the TRS R. Interpretation over the domain with elements from 0 to 1.U52: 0
ISNATILISTKIND: 0
n__isNat: 0
U41: 0
activate: x₀
n__nil: 1
n__s: 0
and: x₁
isNatIListKind: 0
U21: 0
tt: 0
isNatList: 0
AND: 0
zeros: 0
isNatIList: 0
U12: 0
s: 0
isNat: 0
U51: 0
nil: 1
ACTIVATE: 0
U31: 0
n__length: x₀
n__isNatKind: 0
ISNATKIND: 0
U42: 0
n__zeros: 0
n__cons: 0
isNatKind: 0
U11: 0
0: 1
U43: 0
n__and: x₁
cons: 0
U22: 0
U53: 0
U61: 0
n__0: 1
U32: 0
n__isNatIListKind: 0
length: x₀
By semantic labelling [33] we obtain the following labelled TRS:Q DP problem:
The TRS P consists of the following rules:

ACTIVATE.0(n__and.1-0(X1, X2)) → ACTIVATE.1(X1)
ISNATKIND.0(n__s.0(n__s.0(x0))) → ISNATKIND.0(s.0(activate.0(x0)))
ISNATILISTKIND.0(n__cons.1-0(n__and.0-1(x0, x1), y1)) → AND.0-0(isNatKind.1(and.0-1(activate.0(x0), x1)), n__isNatIListKind.0(activate.0(y1)))
ISNATILISTKIND.0(n__cons.1-1(n__0., y1)) → AND.0-0(isNatKind.1(0.), n__isNatIListKind.1(activate.1(y1)))
ISNATKIND.0(n__s.0(n__and.0-0(x0, x1))) → ISNATKIND.0(and.0-0(activate.0(x0), x1))
ISNATKIND.0(n__s.1(n__length.1(x0))) → ISNATKIND.1(length.1(activate.1(x0)))
ISNATILISTKIND.0(n__cons.0-1(n__and.1-0(x0, x1), y1)) → AND.0-0(isNatKind.0(and.1-0(activate.1(x0), x1)), n__isNatIListKind.1(activate.1(y1)))
ISNATKIND.0(n__s.1(x0)) → ISNATKIND.1(x0)
ACTIVATE.0(n__and.0-0(n__isNatKind.0(x0), y1)) → AND.0-0(isNatKind.0(x0), y1)
ACTIVATE.0(n__s.1(X)) → ACTIVATE.1(X)
ISNATKIND.0(n__s.1(n__and.0-1(x0, x1))) → ISNATKIND.1(and.0-1(activate.0(x0), x1))
ISNATKIND.0(n__s.1(V1)) → ACTIVATE.1(V1)
ISNATILISTKIND.0(n__cons.1-0(n__and.1-1(x0, x1), y1)) → AND.0-0(isNatKind.1(and.1-1(activate.1(x0), x1)), n__isNatIListKind.0(activate.0(y1)))
ISNATILISTKIND.0(n__cons.0-0(n__and.0-0(x0, x1), y1)) → AND.0-0(isNatKind.0(and.0-0(activate.0(x0), x1)), n__isNatIListKind.0(activate.0(y1)))
ACTIVATE.0(n__and.0-0(n__and.1-0(x0, x1), y1)) → AND.0-0(and.1-0(activate.1(x0), x1), y1)
ACTIVATE.0(n__and.0-0(x0, y1)) → AND.0-0(x0, y1)
ISNATILISTKIND.0(n__cons.1-1(n__and.0-1(x0, x1), y1)) → AND.0-0(isNatKind.1(and.0-1(activate.0(x0), x1)), n__isNatIListKind.1(activate.1(y1)))
ISNATKIND.0(n__s.0(n__length.0(x0))) → ISNATKIND.0(length.0(activate.0(x0)))
ACTIVATE.1(n__and.1-1(X1, X2)) → ACTIVATE.1(X1)
ISNATKIND.0(n__s.1(n__and.1-1(x0, x1))) → ISNATKIND.1(and.1-1(activate.1(x0), x1))
ACTIVATE.0(n__and.1-0(n__and.1-1(x0, x1), y1)) → AND.1-0(and.1-1(activate.1(x0), x1), y1)
ISNATILISTKIND.0(n__cons.0-0(x0, y1)) → AND.0-0(isNatKind.0(x0), n__isNatIListKind.0(activate.0(y1)))
ACTIVATE.1(n__and.1-1(n__and.1-1(x0, x1), y1)) → AND.1-1(and.1-1(activate.1(x0), x1), y1)
ISNATILISTKIND.0(n__cons.0-1(n__and.0-0(x0, x1), y1)) → AND.0-0(isNatKind.0(and.0-0(activate.0(x0), x1)), n__isNatIListKind.1(activate.1(y1)))
ACTIVATE.1(n__and.0-1(n__isNatKind.1(x0), y1)) → AND.0-1(isNatKind.1(x0), y1)
ISNATKIND.0(n__s.0(n__and.1-0(x0, x1))) → ISNATKIND.0(and.1-0(activate.1(x0), x1))
ACTIVATE.0(n__and.1-0(x0, y1)) → AND.1-0(x0, y1)
ACTIVATE.0(n__and.0-0(X1, X2)) → ACTIVATE.0(X1)
ACTIVATE.0(n__and.1-0(n__and.0-1(x0, x1), y1)) → AND.1-0(and.0-1(activate.0(x0), x1), y1)
ACTIVATE.1(n__and.1-1(n__and.0-1(x0, x1), y1)) → AND.1-1(and.0-1(activate.0(x0), x1), y1)
AND.0-1(tt., X) → ACTIVATE.1(X)
ACTIVATE.0(n__isNatIListKind.0(X)) → ISNATILISTKIND.0(X)
AND.0-0(tt., X) → ACTIVATE.0(X)
ISNATILISTKIND.0(n__cons.1-0(n__0., y1)) → AND.0-0(isNatKind.1(0.), n__isNatIListKind.0(activate.0(y1)))
ACTIVATE.0(n__isNatIListKind.1(X)) → ISNATILISTKIND.1(X)
ACTIVATE.1(n__and.0-1(X1, X2)) → ACTIVATE.0(X1)
ACTIVATE.0(n__and.0-0(n__isNatKind.1(x0), y1)) → AND.0-0(isNatKind.1(x0), y1)
ACTIVATE.0(n__s.0(X)) → ACTIVATE.0(X)
ISNATKIND.0(n__s.0(x0)) → ISNATKIND.0(x0)
ISNATILISTKIND.0(n__cons.0-0(n__and.1-0(x0, x1), y1)) → AND.0-0(isNatKind.0(and.1-0(activate.1(x0), x1)), n__isNatIListKind.0(activate.0(y1)))
ACTIVATE.0(n__and.0-0(n__and.0-0(x0, x1), y1)) → AND.0-0(and.0-0(activate.0(x0), x1), y1)
ISNATILISTKIND.0(n__cons.1-1(x0, y1)) → AND.0-0(isNatKind.1(x0), n__isNatIListKind.1(activate.1(y1)))
ACTIVATE.1(n__and.0-1(n__and.1-0(x0, x1), y1)) → AND.0-1(and.1-0(activate.1(x0), x1), y1)
ACTIVATE.0(n__isNatKind.1(X)) → ISNATKIND.1(X)
ISNATILISTKIND.0(n__cons.1-0(x0, y1)) → AND.0-0(isNatKind.1(x0), n__isNatIListKind.0(activate.0(y1)))
ISNATILISTKIND.0(n__cons.1-1(n__and.1-1(x0, x1), y1)) → AND.0-0(isNatKind.1(and.1-1(activate.1(x0), x1)), n__isNatIListKind.1(activate.1(y1)))
ACTIVATE.0(n__isNatKind.0(X)) → ISNATKIND.0(X)
ACTIVATE.1(n__and.0-1(n__isNatKind.0(x0), y1)) → AND.0-1(isNatKind.0(x0), y1)
ACTIVATE.1(n__and.0-1(x0, y1)) → AND.0-1(x0, y1)
ACTIVATE.1(n__and.1-1(x0, y1)) → AND.1-1(x0, y1)
ACTIVATE.1(n__and.0-1(n__and.0-0(x0, x1), y1)) → AND.0-1(and.0-0(activate.0(x0), x1), y1)
ISNATILISTKIND.0(n__cons.0-1(x0, y1)) → AND.0-0(isNatKind.0(x0), n__isNatIListKind.1(activate.1(y1)))
ISNATKIND.0(n__s.0(V1)) → ACTIVATE.0(V1)
ISNATKIND.0(n__s.0(n__s.1(x0))) → ISNATKIND.0(s.1(activate.1(x0)))

The TRS R consists of the following rules:

U12.0(tt.) → tt.
U61.0-1(tt., L) → s.1(length.1(activate.1(L)))
length.0(cons.0-1(N, L)) → U61.0-1(and.0-0(and.0-0(isNatList.1(activate.1(L)), n__isNatIListKind.1(activate.1(L))), n__and.0-0(n__isNat.0(N), n__isNatKind.0(N))), activate.1(L))
U53.0(tt.) → tt.
activate.0(n__and.1-0(X1, X2)) → and.1-0(activate.1(X1), X2)
activate.1(n__0.) → 0.
isNatIList.0(n__cons.0-1(V1, V2)) → U41.0-0-1(and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.1(activate.1(V2))), activate.0(V1), activate.1(V2))
isNatIList.1(V) → U31.0-1(isNatIListKind.1(activate.1(V)), activate.1(V))
activate.0(n__zeros.) → zeros.
and.0-1(tt., X) → activate.1(X)
isNatKind.1(n__length.1(V1)) → isNatIListKind.1(activate.1(V1))
and.0-0(X1, X2) → n__and.0-0(X1, X2)
activate.0(n__s.0(X)) → s.0(activate.0(X))
isNat.0(n__s.0(V1)) → U21.0-0(isNatKind.0(activate.0(V1)), activate.0(V1))
U51.0-0-1(tt., V1, V2) → U52.0-1(isNat.0(activate.0(V1)), activate.1(V2))
U21.0-1(tt., V1) → U22.0(isNat.1(activate.1(V1)))
isNatList.0(n__cons.1-1(V1, V2)) → U51.0-1-1(and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.1(activate.1(V2))), activate.1(V1), activate.1(V2))
length.0(X) → n__length.0(X)
isNatList.0(n__cons.0-1(V1, V2)) → U51.0-0-1(and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.1(activate.1(V2))), activate.0(V1), activate.1(V2))
U52.0-1(tt., V2) → U53.0(isNatList.1(activate.1(V2)))
cons.1-0(X1, X2) → n__cons.1-0(X1, X2)
length.1(nil.) → 0.
isNatIList.0(n__cons.1-1(V1, V2)) → U41.0-1-1(and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.1(activate.1(V2))), activate.1(V1), activate.1(V2))
and.1-1(X1, X2) → n__and.1-1(X1, X2)
activate.0(n__isNatKind.0(X)) → isNatKind.0(X)
activate.0(n__s.1(X)) → s.1(activate.1(X))
U31.0-0(tt., V) → U32.0(isNatList.0(activate.0(V)))
isNatIListKind.0(n__cons.0-0(V1, V2)) → and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.0(activate.0(V2)))
isNat.0(X) → n__isNat.0(X)
length.0(cons.0-0(N, L)) → U61.0-0(and.0-0(and.0-0(isNatList.0(activate.0(L)), n__isNatIListKind.0(activate.0(L))), n__and.0-0(n__isNat.0(N), n__isNatKind.0(N))), activate.0(L))
activate.0(n__and.0-0(X1, X2)) → and.0-0(activate.0(X1), X2)
zeros. → cons.1-0(0., n__zeros.)
activate.0(n__isNat.1(X)) → isNat.1(X)
activate.0(n__isNatIListKind.0(X)) → isNatIListKind.0(X)
cons.0-0(X1, X2) → n__cons.0-0(X1, X2)
isNatIListKind.0(n__cons.1-1(V1, V2)) → and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.1(activate.1(V2)))
s.0(X) → n__s.0(X)
cons.0-1(X1, X2) → n__cons.0-1(X1, X2)
U41.0-1-0(tt., V1, V2) → U42.0-0(isNat.1(activate.1(V1)), activate.0(V2))
isNatKind.0(X) → n__isNatKind.0(X)
activate.0(n__cons.0-1(X1, X2)) → cons.0-1(activate.0(X1), X2)
isNat.1(n__length.1(V1)) → U11.0-1(isNatIListKind.1(activate.1(V1)), activate.1(V1))
length.0(cons.1-1(N, L)) → U61.0-1(and.0-0(and.0-0(isNatList.1(activate.1(L)), n__isNatIListKind.1(activate.1(L))), n__and.0-0(n__isNat.1(N), n__isNatKind.1(N))), activate.1(L))
isNat.1(n__0.) → tt.
and.0-1(X1, X2) → n__and.0-1(X1, X2)
U21.0-0(tt., V1) → U22.0(isNat.0(activate.0(V1)))
isNatKind.0(n__length.0(V1)) → isNatIListKind.0(activate.0(V1))
zeros. → n__zeros.
U11.0-1(tt., V1) → U12.0(isNatList.1(activate.1(V1)))
U11.0-0(tt., V1) → U12.0(isNatList.0(activate.0(V1)))
U43.0(tt.) → tt.
activate.0(n__isNatKind.1(X)) → isNatKind.1(X)
isNat.1(X) → n__isNat.1(X)
activate.1(n__and.1-1(X1, X2)) → and.1-1(activate.1(X1), X2)
U31.0-1(tt., V) → U32.0(isNatList.1(activate.1(V)))
activate.0(n__cons.0-0(X1, X2)) → cons.0-0(activate.0(X1), X2)
U41.0-0-0(tt., V1, V2) → U42.0-0(isNat.0(activate.0(V1)), activate.0(V2))
U32.0(tt.) → tt.
isNat.0(n__length.0(V1)) → U11.0-0(isNatIListKind.0(activate.0(V1)), activate.0(V1))
isNatIListKind.0(X) → n__isNatIListKind.0(X)
isNat.0(n__s.1(V1)) → U21.0-1(isNatKind.1(activate.1(V1)), activate.1(V1))
s.1(X) → n__s.1(X)
U22.0(tt.) → tt.
isNatIListKind.1(X) → n__isNatIListKind.1(X)
U51.0-0-0(tt., V1, V2) → U52.0-0(isNat.0(activate.0(V1)), activate.0(V2))
isNatIListKind.0(n__zeros.) → tt.
activate.1(n__nil.) → nil.
isNatKind.0(n__s.0(V1)) → isNatKind.0(activate.0(V1))
isNatIList.0(n__cons.0-0(V1, V2)) → U41.0-0-0(and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.0(activate.0(V2))), activate.0(V1), activate.0(V2))
activate.0(n__cons.1-1(X1, X2)) → cons.1-1(activate.1(X1), X2)
isNatIList.0(n__cons.1-0(V1, V2)) → U41.0-1-0(and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.0(activate.0(V2))), activate.1(V1), activate.0(V2))
and.0-0(tt., X) → activate.0(X)
U42.0-1(tt., V2) → U43.0(isNatIList.1(activate.1(V2)))
length.1(X) → n__length.1(X)
cons.1-1(X1, X2) → n__cons.1-1(X1, X2)
and.1-0(X1, X2) → n__and.1-0(X1, X2)
activate.0(X) → X
U61.0-0(tt., L) → s.0(length.0(activate.0(L)))
activate.0(n__isNat.0(X)) → isNat.0(X)
activate.1(X) → X
isNatIListKind.1(n__nil.) → tt.
isNatKind.1(n__0.) → tt.
0. → n__0.
U41.0-1-1(tt., V1, V2) → U42.0-1(isNat.1(activate.1(V1)), activate.1(V2))
length.0(cons.1-0(N, L)) → U61.0-0(and.0-0(and.0-0(isNatList.0(activate.0(L)), n__isNatIListKind.0(activate.0(L))), n__and.0-0(n__isNat.1(N), n__isNatKind.1(N))), activate.0(L))
isNatKind.1(X) → n__isNatKind.1(X)
isNatIListKind.0(n__cons.1-0(V1, V2)) → and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.0(activate.0(V2)))
isNatIListKind.0(n__cons.0-1(V1, V2)) → and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.1(activate.1(V2)))
isNatIList.0(V) → U31.0-0(isNatIListKind.0(activate.0(V)), activate.0(V))
U42.0-0(tt., V2) → U43.0(isNatIList.0(activate.0(V2)))
isNatList.0(n__cons.1-0(V1, V2)) → U51.0-1-0(and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.0(activate.0(V2))), activate.1(V1), activate.0(V2))
activate.0(n__cons.1-0(X1, X2)) → cons.1-0(activate.1(X1), X2)
activate.1(n__length.1(X)) → length.1(activate.1(X))
isNatIList.0(n__zeros.) → tt.
activate.0(n__length.0(X)) → length.0(activate.0(X))
activate.1(n__and.0-1(X1, X2)) → and.0-1(activate.0(X1), X2)
U51.0-1-1(tt., V1, V2) → U52.0-1(isNat.1(activate.1(V1)), activate.1(V2))
U41.0-0-1(tt., V1, V2) → U42.0-1(isNat.0(activate.0(V1)), activate.1(V2))
U52.0-0(tt., V2) → U53.0(isNatList.0(activate.0(V2)))
U51.0-1-0(tt., V1, V2) → U52.0-0(isNat.1(activate.1(V1)), activate.0(V2))
isNatKind.0(n__s.1(V1)) → isNatKind.1(activate.1(V1))
isNatList.0(n__cons.0-0(V1, V2)) → U51.0-0-0(and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.0(activate.0(V2))), activate.0(V1), activate.0(V2))
nil. → n__nil.
isNatList.1(n__nil.) → tt.
activate.0(n__isNatIListKind.1(X)) → isNatIListKind.1(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

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ 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

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ SemLabProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                            ↳ SemLabProof2

                              ↳ QDP

                    ↳ QDP

          ↳ QDP

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

ACTIVATE.0(n__and.1-0(X1, X2)) → ACTIVATE.1(X1)
ISNATKIND.0(n__s.0(n__s.0(x0))) → ISNATKIND.0(s.0(activate.0(x0)))
ISNATILISTKIND.0(n__cons.1-0(n__and.0-1(x0, x1), y1)) → AND.0-0(isNatKind.1(and.0-1(activate.0(x0), x1)), n__isNatIListKind.0(activate.0(y1)))
ISNATILISTKIND.0(n__cons.1-1(n__0., y1)) → AND.0-0(isNatKind.1(0.), n__isNatIListKind.1(activate.1(y1)))
ISNATKIND.0(n__s.0(n__and.0-0(x0, x1))) → ISNATKIND.0(and.0-0(activate.0(x0), x1))
ISNATKIND.0(n__s.1(n__length.1(x0))) → ISNATKIND.1(length.1(activate.1(x0)))
ISNATILISTKIND.0(n__cons.0-1(n__and.1-0(x0, x1), y1)) → AND.0-0(isNatKind.0(and.1-0(activate.1(x0), x1)), n__isNatIListKind.1(activate.1(y1)))
ISNATKIND.0(n__s.1(x0)) → ISNATKIND.1(x0)
ACTIVATE.0(n__and.0-0(n__isNatKind.0(x0), y1)) → AND.0-0(isNatKind.0(x0), y1)
ACTIVATE.0(n__s.1(X)) → ACTIVATE.1(X)
ISNATKIND.0(n__s.1(n__and.0-1(x0, x1))) → ISNATKIND.1(and.0-1(activate.0(x0), x1))
ISNATKIND.0(n__s.1(V1)) → ACTIVATE.1(V1)
ISNATILISTKIND.0(n__cons.1-0(n__and.1-1(x0, x1), y1)) → AND.0-0(isNatKind.1(and.1-1(activate.1(x0), x1)), n__isNatIListKind.0(activate.0(y1)))
ISNATILISTKIND.0(n__cons.0-0(n__and.0-0(x0, x1), y1)) → AND.0-0(isNatKind.0(and.0-0(activate.0(x0), x1)), n__isNatIListKind.0(activate.0(y1)))
ACTIVATE.0(n__and.0-0(n__and.1-0(x0, x1), y1)) → AND.0-0(and.1-0(activate.1(x0), x1), y1)
ACTIVATE.0(n__and.0-0(x0, y1)) → AND.0-0(x0, y1)
ISNATILISTKIND.0(n__cons.1-1(n__and.0-1(x0, x1), y1)) → AND.0-0(isNatKind.1(and.0-1(activate.0(x0), x1)), n__isNatIListKind.1(activate.1(y1)))
ISNATKIND.0(n__s.0(n__length.0(x0))) → ISNATKIND.0(length.0(activate.0(x0)))
ACTIVATE.1(n__and.1-1(X1, X2)) → ACTIVATE.1(X1)
ISNATKIND.0(n__s.1(n__and.1-1(x0, x1))) → ISNATKIND.1(and.1-1(activate.1(x0), x1))
ACTIVATE.0(n__and.1-0(n__and.1-1(x0, x1), y1)) → AND.1-0(and.1-1(activate.1(x0), x1), y1)
ISNATILISTKIND.0(n__cons.0-0(x0, y1)) → AND.0-0(isNatKind.0(x0), n__isNatIListKind.0(activate.0(y1)))
ACTIVATE.1(n__and.1-1(n__and.1-1(x0, x1), y1)) → AND.1-1(and.1-1(activate.1(x0), x1), y1)
ISNATILISTKIND.0(n__cons.0-1(n__and.0-0(x0, x1), y1)) → AND.0-0(isNatKind.0(and.0-0(activate.0(x0), x1)), n__isNatIListKind.1(activate.1(y1)))
ACTIVATE.1(n__and.0-1(n__isNatKind.1(x0), y1)) → AND.0-1(isNatKind.1(x0), y1)
ISNATKIND.0(n__s.0(n__and.1-0(x0, x1))) → ISNATKIND.0(and.1-0(activate.1(x0), x1))
ACTIVATE.0(n__and.1-0(x0, y1)) → AND.1-0(x0, y1)
ACTIVATE.0(n__and.0-0(X1, X2)) → ACTIVATE.0(X1)
ACTIVATE.0(n__and.1-0(n__and.0-1(x0, x1), y1)) → AND.1-0(and.0-1(activate.0(x0), x1), y1)
ACTIVATE.1(n__and.1-1(n__and.0-1(x0, x1), y1)) → AND.1-1(and.0-1(activate.0(x0), x1), y1)
AND.0-1(tt., X) → ACTIVATE.1(X)
ACTIVATE.0(n__isNatIListKind.0(X)) → ISNATILISTKIND.0(X)
AND.0-0(tt., X) → ACTIVATE.0(X)
ISNATILISTKIND.0(n__cons.1-0(n__0., y1)) → AND.0-0(isNatKind.1(0.), n__isNatIListKind.0(activate.0(y1)))
ACTIVATE.0(n__isNatIListKind.1(X)) → ISNATILISTKIND.1(X)
ACTIVATE.1(n__and.0-1(X1, X2)) → ACTIVATE.0(X1)
ACTIVATE.0(n__and.0-0(n__isNatKind.1(x0), y1)) → AND.0-0(isNatKind.1(x0), y1)
ACTIVATE.0(n__s.0(X)) → ACTIVATE.0(X)
ISNATKIND.0(n__s.0(x0)) → ISNATKIND.0(x0)
ISNATILISTKIND.0(n__cons.0-0(n__and.1-0(x0, x1), y1)) → AND.0-0(isNatKind.0(and.1-0(activate.1(x0), x1)), n__isNatIListKind.0(activate.0(y1)))
ACTIVATE.0(n__and.0-0(n__and.0-0(x0, x1), y1)) → AND.0-0(and.0-0(activate.0(x0), x1), y1)
ISNATILISTKIND.0(n__cons.1-1(x0, y1)) → AND.0-0(isNatKind.1(x0), n__isNatIListKind.1(activate.1(y1)))
ACTIVATE.1(n__and.0-1(n__and.1-0(x0, x1), y1)) → AND.0-1(and.1-0(activate.1(x0), x1), y1)
ACTIVATE.0(n__isNatKind.1(X)) → ISNATKIND.1(X)
ISNATILISTKIND.0(n__cons.1-0(x0, y1)) → AND.0-0(isNatKind.1(x0), n__isNatIListKind.0(activate.0(y1)))
ISNATILISTKIND.0(n__cons.1-1(n__and.1-1(x0, x1), y1)) → AND.0-0(isNatKind.1(and.1-1(activate.1(x0), x1)), n__isNatIListKind.1(activate.1(y1)))
ACTIVATE.0(n__isNatKind.0(X)) → ISNATKIND.0(X)
ACTIVATE.1(n__and.0-1(n__isNatKind.0(x0), y1)) → AND.0-1(isNatKind.0(x0), y1)
ACTIVATE.1(n__and.0-1(x0, y1)) → AND.0-1(x0, y1)
ACTIVATE.1(n__and.1-1(x0, y1)) → AND.1-1(x0, y1)
ACTIVATE.1(n__and.0-1(n__and.0-0(x0, x1), y1)) → AND.0-1(and.0-0(activate.0(x0), x1), y1)
ISNATILISTKIND.0(n__cons.0-1(x0, y1)) → AND.0-0(isNatKind.0(x0), n__isNatIListKind.1(activate.1(y1)))
ISNATKIND.0(n__s.0(V1)) → ACTIVATE.0(V1)
ISNATKIND.0(n__s.0(n__s.1(x0))) → ISNATKIND.0(s.1(activate.1(x0)))

The TRS R consists of the following rules:

U12.0(tt.) → tt.
U61.0-1(tt., L) → s.1(length.1(activate.1(L)))
length.0(cons.0-1(N, L)) → U61.0-1(and.0-0(and.0-0(isNatList.1(activate.1(L)), n__isNatIListKind.1(activate.1(L))), n__and.0-0(n__isNat.0(N), n__isNatKind.0(N))), activate.1(L))
U53.0(tt.) → tt.
activate.0(n__and.1-0(X1, X2)) → and.1-0(activate.1(X1), X2)
activate.1(n__0.) → 0.
isNatIList.0(n__cons.0-1(V1, V2)) → U41.0-0-1(and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.1(activate.1(V2))), activate.0(V1), activate.1(V2))
isNatIList.1(V) → U31.0-1(isNatIListKind.1(activate.1(V)), activate.1(V))
activate.0(n__zeros.) → zeros.
and.0-1(tt., X) → activate.1(X)
isNatKind.1(n__length.1(V1)) → isNatIListKind.1(activate.1(V1))
and.0-0(X1, X2) → n__and.0-0(X1, X2)
activate.0(n__s.0(X)) → s.0(activate.0(X))
isNat.0(n__s.0(V1)) → U21.0-0(isNatKind.0(activate.0(V1)), activate.0(V1))
U51.0-0-1(tt., V1, V2) → U52.0-1(isNat.0(activate.0(V1)), activate.1(V2))
U21.0-1(tt., V1) → U22.0(isNat.1(activate.1(V1)))
isNatList.0(n__cons.1-1(V1, V2)) → U51.0-1-1(and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.1(activate.1(V2))), activate.1(V1), activate.1(V2))
length.0(X) → n__length.0(X)
isNatList.0(n__cons.0-1(V1, V2)) → U51.0-0-1(and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.1(activate.1(V2))), activate.0(V1), activate.1(V2))
U52.0-1(tt., V2) → U53.0(isNatList.1(activate.1(V2)))
cons.1-0(X1, X2) → n__cons.1-0(X1, X2)
length.1(nil.) → 0.
isNatIList.0(n__cons.1-1(V1, V2)) → U41.0-1-1(and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.1(activate.1(V2))), activate.1(V1), activate.1(V2))
and.1-1(X1, X2) → n__and.1-1(X1, X2)
activate.0(n__isNatKind.0(X)) → isNatKind.0(X)
activate.0(n__s.1(X)) → s.1(activate.1(X))
U31.0-0(tt., V) → U32.0(isNatList.0(activate.0(V)))
isNatIListKind.0(n__cons.0-0(V1, V2)) → and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.0(activate.0(V2)))
isNat.0(X) → n__isNat.0(X)
length.0(cons.0-0(N, L)) → U61.0-0(and.0-0(and.0-0(isNatList.0(activate.0(L)), n__isNatIListKind.0(activate.0(L))), n__and.0-0(n__isNat.0(N), n__isNatKind.0(N))), activate.0(L))
activate.0(n__and.0-0(X1, X2)) → and.0-0(activate.0(X1), X2)
zeros. → cons.1-0(0., n__zeros.)
activate.0(n__isNat.1(X)) → isNat.1(X)
activate.0(n__isNatIListKind.0(X)) → isNatIListKind.0(X)
cons.0-0(X1, X2) → n__cons.0-0(X1, X2)
isNatIListKind.0(n__cons.1-1(V1, V2)) → and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.1(activate.1(V2)))
s.0(X) → n__s.0(X)
cons.0-1(X1, X2) → n__cons.0-1(X1, X2)
U41.0-1-0(tt., V1, V2) → U42.0-0(isNat.1(activate.1(V1)), activate.0(V2))
isNatKind.0(X) → n__isNatKind.0(X)
activate.0(n__cons.0-1(X1, X2)) → cons.0-1(activate.0(X1), X2)
isNat.1(n__length.1(V1)) → U11.0-1(isNatIListKind.1(activate.1(V1)), activate.1(V1))
length.0(cons.1-1(N, L)) → U61.0-1(and.0-0(and.0-0(isNatList.1(activate.1(L)), n__isNatIListKind.1(activate.1(L))), n__and.0-0(n__isNat.1(N), n__isNatKind.1(N))), activate.1(L))
isNat.1(n__0.) → tt.
and.0-1(X1, X2) → n__and.0-1(X1, X2)
U21.0-0(tt., V1) → U22.0(isNat.0(activate.0(V1)))
isNatKind.0(n__length.0(V1)) → isNatIListKind.0(activate.0(V1))
zeros. → n__zeros.
U11.0-1(tt., V1) → U12.0(isNatList.1(activate.1(V1)))
U11.0-0(tt., V1) → U12.0(isNatList.0(activate.0(V1)))
U43.0(tt.) → tt.
activate.0(n__isNatKind.1(X)) → isNatKind.1(X)
isNat.1(X) → n__isNat.1(X)
activate.1(n__and.1-1(X1, X2)) → and.1-1(activate.1(X1), X2)
U31.0-1(tt., V) → U32.0(isNatList.1(activate.1(V)))
activate.0(n__cons.0-0(X1, X2)) → cons.0-0(activate.0(X1), X2)
U41.0-0-0(tt., V1, V2) → U42.0-0(isNat.0(activate.0(V1)), activate.0(V2))
U32.0(tt.) → tt.
isNat.0(n__length.0(V1)) → U11.0-0(isNatIListKind.0(activate.0(V1)), activate.0(V1))
isNatIListKind.0(X) → n__isNatIListKind.0(X)
isNat.0(n__s.1(V1)) → U21.0-1(isNatKind.1(activate.1(V1)), activate.1(V1))
s.1(X) → n__s.1(X)
U22.0(tt.) → tt.
isNatIListKind.1(X) → n__isNatIListKind.1(X)
U51.0-0-0(tt., V1, V2) → U52.0-0(isNat.0(activate.0(V1)), activate.0(V2))
isNatIListKind.0(n__zeros.) → tt.
activate.1(n__nil.) → nil.
isNatKind.0(n__s.0(V1)) → isNatKind.0(activate.0(V1))
isNatIList.0(n__cons.0-0(V1, V2)) → U41.0-0-0(and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.0(activate.0(V2))), activate.0(V1), activate.0(V2))
activate.0(n__cons.1-1(X1, X2)) → cons.1-1(activate.1(X1), X2)
isNatIList.0(n__cons.1-0(V1, V2)) → U41.0-1-0(and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.0(activate.0(V2))), activate.1(V1), activate.0(V2))
and.0-0(tt., X) → activate.0(X)
U42.0-1(tt., V2) → U43.0(isNatIList.1(activate.1(V2)))
length.1(X) → n__length.1(X)
cons.1-1(X1, X2) → n__cons.1-1(X1, X2)
and.1-0(X1, X2) → n__and.1-0(X1, X2)
activate.0(X) → X
U61.0-0(tt., L) → s.0(length.0(activate.0(L)))
activate.0(n__isNat.0(X)) → isNat.0(X)
activate.1(X) → X
isNatIListKind.1(n__nil.) → tt.
isNatKind.1(n__0.) → tt.
0. → n__0.
U41.0-1-1(tt., V1, V2) → U42.0-1(isNat.1(activate.1(V1)), activate.1(V2))
length.0(cons.1-0(N, L)) → U61.0-0(and.0-0(and.0-0(isNatList.0(activate.0(L)), n__isNatIListKind.0(activate.0(L))), n__and.0-0(n__isNat.1(N), n__isNatKind.1(N))), activate.0(L))
isNatKind.1(X) → n__isNatKind.1(X)
isNatIListKind.0(n__cons.1-0(V1, V2)) → and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.0(activate.0(V2)))
isNatIListKind.0(n__cons.0-1(V1, V2)) → and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.1(activate.1(V2)))
isNatIList.0(V) → U31.0-0(isNatIListKind.0(activate.0(V)), activate.0(V))
U42.0-0(tt., V2) → U43.0(isNatIList.0(activate.0(V2)))
isNatList.0(n__cons.1-0(V1, V2)) → U51.0-1-0(and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.0(activate.0(V2))), activate.1(V1), activate.0(V2))
activate.0(n__cons.1-0(X1, X2)) → cons.1-0(activate.1(X1), X2)
activate.1(n__length.1(X)) → length.1(activate.1(X))
isNatIList.0(n__zeros.) → tt.
activate.0(n__length.0(X)) → length.0(activate.0(X))
activate.1(n__and.0-1(X1, X2)) → and.0-1(activate.0(X1), X2)
U51.0-1-1(tt., V1, V2) → U52.0-1(isNat.1(activate.1(V1)), activate.1(V2))
U41.0-0-1(tt., V1, V2) → U42.0-1(isNat.0(activate.0(V1)), activate.1(V2))
U52.0-0(tt., V2) → U53.0(isNatList.0(activate.0(V2)))
U51.0-1-0(tt., V1, V2) → U52.0-0(isNat.1(activate.1(V1)), activate.0(V2))
isNatKind.0(n__s.1(V1)) → isNatKind.1(activate.1(V1))
isNatList.0(n__cons.0-0(V1, V2)) → U51.0-0-0(and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.0(activate.0(V2))), activate.0(V1), activate.0(V2))
nil. → n__nil.
isNatList.1(n__nil.) → tt.
activate.0(n__isNatIListKind.1(X)) → isNatIListKind.1(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 12 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ 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

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ SemLabProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                                                            ↳ SemLabProof2

                              ↳ QDP

                    ↳ QDP

          ↳ QDP

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

ACTIVATE.1(n__and.0-1(n__isNatKind.1(x0), y1)) → AND.0-1(isNatKind.1(x0), y1)
ISNATKIND.0(n__s.0(n__and.1-0(x0, x1))) → ISNATKIND.0(and.1-0(activate.1(x0), x1))
ACTIVATE.0(n__and.1-0(X1, X2)) → ACTIVATE.1(X1)
ACTIVATE.0(n__and.0-0(X1, X2)) → ACTIVATE.0(X1)
ISNATKIND.0(n__s.0(n__s.0(x0))) → ISNATKIND.0(s.0(activate.0(x0)))
ISNATILISTKIND.0(n__cons.1-0(n__and.0-1(x0, x1), y1)) → AND.0-0(isNatKind.1(and.0-1(activate.0(x0), x1)), n__isNatIListKind.0(activate.0(y1)))
AND.0-1(tt., X) → ACTIVATE.1(X)
ISNATILISTKIND.0(n__cons.1-1(n__0., y1)) → AND.0-0(isNatKind.1(0.), n__isNatIListKind.1(activate.1(y1)))
ACTIVATE.0(n__isNatIListKind.0(X)) → ISNATILISTKIND.0(X)
ISNATKIND.0(n__s.0(n__and.0-0(x0, x1))) → ISNATKIND.0(and.0-0(activate.0(x0), x1))
AND.0-0(tt., X) → ACTIVATE.0(X)
ISNATILISTKIND.0(n__cons.1-0(n__0., y1)) → AND.0-0(isNatKind.1(0.), n__isNatIListKind.0(activate.0(y1)))
ISNATILISTKIND.0(n__cons.0-1(n__and.1-0(x0, x1), y1)) → AND.0-0(isNatKind.0(and.1-0(activate.1(x0), x1)), n__isNatIListKind.1(activate.1(y1)))
ACTIVATE.0(n__and.0-0(n__isNatKind.0(x0), y1)) → AND.0-0(isNatKind.0(x0), y1)
ACTIVATE.0(n__s.1(X)) → ACTIVATE.1(X)
ACTIVATE.1(n__and.0-1(X1, X2)) → ACTIVATE.0(X1)
ISNATKIND.0(n__s.1(V1)) → ACTIVATE.1(V1)
ACTIVATE.0(n__and.0-0(n__isNatKind.1(x0), y1)) → AND.0-0(isNatKind.1(x0), y1)
ACTIVATE.0(n__s.0(X)) → ACTIVATE.0(X)
ISNATILISTKIND.0(n__cons.1-0(n__and.1-1(x0, x1), y1)) → AND.0-0(isNatKind.1(and.1-1(activate.1(x0), x1)), n__isNatIListKind.0(activate.0(y1)))
ISNATILISTKIND.0(n__cons.0-0(n__and.0-0(x0, x1), y1)) → AND.0-0(isNatKind.0(and.0-0(activate.0(x0), x1)), n__isNatIListKind.0(activate.0(y1)))
ISNATKIND.0(n__s.0(x0)) → ISNATKIND.0(x0)
ACTIVATE.0(n__and.0-0(n__and.1-0(x0, x1), y1)) → AND.0-0(and.1-0(activate.1(x0), x1), y1)
ACTIVATE.0(n__and.0-0(n__and.0-0(x0, x1), y1)) → AND.0-0(and.0-0(activate.0(x0), x1), y1)
ISNATILISTKIND.0(n__cons.0-0(n__and.1-0(x0, x1), y1)) → AND.0-0(isNatKind.0(and.1-0(activate.1(x0), x1)), n__isNatIListKind.0(activate.0(y1)))
ISNATILISTKIND.0(n__cons.1-1(x0, y1)) → AND.0-0(isNatKind.1(x0), n__isNatIListKind.1(activate.1(y1)))
ACTIVATE.1(n__and.0-1(n__and.1-0(x0, x1), y1)) → AND.0-1(and.1-0(activate.1(x0), x1), y1)
ACTIVATE.0(n__and.0-0(x0, y1)) → AND.0-0(x0, y1)
ISNATILISTKIND.0(n__cons.1-1(n__and.0-1(x0, x1), y1)) → AND.0-0(isNatKind.1(and.0-1(activate.0(x0), x1)), n__isNatIListKind.1(activate.1(y1)))
ISNATILISTKIND.0(n__cons.1-0(x0, y1)) → AND.0-0(isNatKind.1(x0), n__isNatIListKind.0(activate.0(y1)))
ISNATKIND.0(n__s.0(n__length.0(x0))) → ISNATKIND.0(length.0(activate.0(x0)))
ACTIVATE.1(n__and.1-1(X1, X2)) → ACTIVATE.1(X1)
ISNATILISTKIND.0(n__cons.1-1(n__and.1-1(x0, x1), y1)) → AND.0-0(isNatKind.1(and.1-1(activate.1(x0), x1)), n__isNatIListKind.1(activate.1(y1)))
ACTIVATE.0(n__isNatKind.0(X)) → ISNATKIND.0(X)
ACTIVATE.1(n__and.0-1(x0, y1)) → AND.0-1(x0, y1)
ISNATILISTKIND.0(n__cons.0-0(x0, y1)) → AND.0-0(isNatKind.0(x0), n__isNatIListKind.0(activate.0(y1)))
ACTIVATE.1(n__and.0-1(n__isNatKind.0(x0), y1)) → AND.0-1(isNatKind.0(x0), y1)
ACTIVATE.1(n__and.0-1(n__and.0-0(x0, x1), y1)) → AND.0-1(and.0-0(activate.0(x0), x1), y1)
ISNATILISTKIND.0(n__cons.0-1(x0, y1)) → AND.0-0(isNatKind.0(x0), n__isNatIListKind.1(activate.1(y1)))
ISNATILISTKIND.0(n__cons.0-1(n__and.0-0(x0, x1), y1)) → AND.0-0(isNatKind.0(and.0-0(activate.0(x0), x1)), n__isNatIListKind.1(activate.1(y1)))
ISNATKIND.0(n__s.0(n__s.1(x0))) → ISNATKIND.0(s.1(activate.1(x0)))
ISNATKIND.0(n__s.0(V1)) → ACTIVATE.0(V1)

The TRS R consists of the following rules:

U12.0(tt.) → tt.
U61.0-1(tt., L) → s.1(length.1(activate.1(L)))
length.0(cons.0-1(N, L)) → U61.0-1(and.0-0(and.0-0(isNatList.1(activate.1(L)), n__isNatIListKind.1(activate.1(L))), n__and.0-0(n__isNat.0(N), n__isNatKind.0(N))), activate.1(L))
U53.0(tt.) → tt.
activate.0(n__and.1-0(X1, X2)) → and.1-0(activate.1(X1), X2)
activate.1(n__0.) → 0.
isNatIList.0(n__cons.0-1(V1, V2)) → U41.0-0-1(and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.1(activate.1(V2))), activate.0(V1), activate.1(V2))
isNatIList.1(V) → U31.0-1(isNatIListKind.1(activate.1(V)), activate.1(V))
activate.0(n__zeros.) → zeros.
and.0-1(tt., X) → activate.1(X)
isNatKind.1(n__length.1(V1)) → isNatIListKind.1(activate.1(V1))
and.0-0(X1, X2) → n__and.0-0(X1, X2)
activate.0(n__s.0(X)) → s.0(activate.0(X))
isNat.0(n__s.0(V1)) → U21.0-0(isNatKind.0(activate.0(V1)), activate.0(V1))
U51.0-0-1(tt., V1, V2) → U52.0-1(isNat.0(activate.0(V1)), activate.1(V2))
U21.0-1(tt., V1) → U22.0(isNat.1(activate.1(V1)))
isNatList.0(n__cons.1-1(V1, V2)) → U51.0-1-1(and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.1(activate.1(V2))), activate.1(V1), activate.1(V2))
length.0(X) → n__length.0(X)
isNatList.0(n__cons.0-1(V1, V2)) → U51.0-0-1(and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.1(activate.1(V2))), activate.0(V1), activate.1(V2))
U52.0-1(tt., V2) → U53.0(isNatList.1(activate.1(V2)))
cons.1-0(X1, X2) → n__cons.1-0(X1, X2)
length.1(nil.) → 0.
isNatIList.0(n__cons.1-1(V1, V2)) → U41.0-1-1(and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.1(activate.1(V2))), activate.1(V1), activate.1(V2))
and.1-1(X1, X2) → n__and.1-1(X1, X2)
activate.0(n__isNatKind.0(X)) → isNatKind.0(X)
activate.0(n__s.1(X)) → s.1(activate.1(X))
U31.0-0(tt., V) → U32.0(isNatList.0(activate.0(V)))
isNatIListKind.0(n__cons.0-0(V1, V2)) → and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.0(activate.0(V2)))
isNat.0(X) → n__isNat.0(X)
length.0(cons.0-0(N, L)) → U61.0-0(and.0-0(and.0-0(isNatList.0(activate.0(L)), n__isNatIListKind.0(activate.0(L))), n__and.0-0(n__isNat.0(N), n__isNatKind.0(N))), activate.0(L))
activate.0(n__and.0-0(X1, X2)) → and.0-0(activate.0(X1), X2)
zeros. → cons.1-0(0., n__zeros.)
activate.0(n__isNat.1(X)) → isNat.1(X)
activate.0(n__isNatIListKind.0(X)) → isNatIListKind.0(X)
cons.0-0(X1, X2) → n__cons.0-0(X1, X2)
isNatIListKind.0(n__cons.1-1(V1, V2)) → and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.1(activate.1(V2)))
s.0(X) → n__s.0(X)
cons.0-1(X1, X2) → n__cons.0-1(X1, X2)
U41.0-1-0(tt., V1, V2) → U42.0-0(isNat.1(activate.1(V1)), activate.0(V2))
isNatKind.0(X) → n__isNatKind.0(X)
activate.0(n__cons.0-1(X1, X2)) → cons.0-1(activate.0(X1), X2)
isNat.1(n__length.1(V1)) → U11.0-1(isNatIListKind.1(activate.1(V1)), activate.1(V1))
length.0(cons.1-1(N, L)) → U61.0-1(and.0-0(and.0-0(isNatList.1(activate.1(L)), n__isNatIListKind.1(activate.1(L))), n__and.0-0(n__isNat.1(N), n__isNatKind.1(N))), activate.1(L))
isNat.1(n__0.) → tt.
and.0-1(X1, X2) → n__and.0-1(X1, X2)
U21.0-0(tt., V1) → U22.0(isNat.0(activate.0(V1)))
isNatKind.0(n__length.0(V1)) → isNatIListKind.0(activate.0(V1))
zeros. → n__zeros.
U11.0-1(tt., V1) → U12.0(isNatList.1(activate.1(V1)))
U11.0-0(tt., V1) → U12.0(isNatList.0(activate.0(V1)))
U43.0(tt.) → tt.
activate.0(n__isNatKind.1(X)) → isNatKind.1(X)
isNat.1(X) → n__isNat.1(X)
activate.1(n__and.1-1(X1, X2)) → and.1-1(activate.1(X1), X2)
U31.0-1(tt., V) → U32.0(isNatList.1(activate.1(V)))
activate.0(n__cons.0-0(X1, X2)) → cons.0-0(activate.0(X1), X2)
U41.0-0-0(tt., V1, V2) → U42.0-0(isNat.0(activate.0(V1)), activate.0(V2))
U32.0(tt.) → tt.
isNat.0(n__length.0(V1)) → U11.0-0(isNatIListKind.0(activate.0(V1)), activate.0(V1))
isNatIListKind.0(X) → n__isNatIListKind.0(X)
isNat.0(n__s.1(V1)) → U21.0-1(isNatKind.1(activate.1(V1)), activate.1(V1))
s.1(X) → n__s.1(X)
U22.0(tt.) → tt.
isNatIListKind.1(X) → n__isNatIListKind.1(X)
U51.0-0-0(tt., V1, V2) → U52.0-0(isNat.0(activate.0(V1)), activate.0(V2))
isNatIListKind.0(n__zeros.) → tt.
activate.1(n__nil.) → nil.
isNatKind.0(n__s.0(V1)) → isNatKind.0(activate.0(V1))
isNatIList.0(n__cons.0-0(V1, V2)) → U41.0-0-0(and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.0(activate.0(V2))), activate.0(V1), activate.0(V2))
activate.0(n__cons.1-1(X1, X2)) → cons.1-1(activate.1(X1), X2)
isNatIList.0(n__cons.1-0(V1, V2)) → U41.0-1-0(and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.0(activate.0(V2))), activate.1(V1), activate.0(V2))
and.0-0(tt., X) → activate.0(X)
U42.0-1(tt., V2) → U43.0(isNatIList.1(activate.1(V2)))
length.1(X) → n__length.1(X)
cons.1-1(X1, X2) → n__cons.1-1(X1, X2)
and.1-0(X1, X2) → n__and.1-0(X1, X2)
activate.0(X) → X
U61.0-0(tt., L) → s.0(length.0(activate.0(L)))
activate.0(n__isNat.0(X)) → isNat.0(X)
activate.1(X) → X
isNatIListKind.1(n__nil.) → tt.
isNatKind.1(n__0.) → tt.
0. → n__0.
U41.0-1-1(tt., V1, V2) → U42.0-1(isNat.1(activate.1(V1)), activate.1(V2))
length.0(cons.1-0(N, L)) → U61.0-0(and.0-0(and.0-0(isNatList.0(activate.0(L)), n__isNatIListKind.0(activate.0(L))), n__and.0-0(n__isNat.1(N), n__isNatKind.1(N))), activate.0(L))
isNatKind.1(X) → n__isNatKind.1(X)
isNatIListKind.0(n__cons.1-0(V1, V2)) → and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.0(activate.0(V2)))
isNatIListKind.0(n__cons.0-1(V1, V2)) → and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.1(activate.1(V2)))
isNatIList.0(V) → U31.0-0(isNatIListKind.0(activate.0(V)), activate.0(V))
U42.0-0(tt., V2) → U43.0(isNatIList.0(activate.0(V2)))
isNatList.0(n__cons.1-0(V1, V2)) → U51.0-1-0(and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.0(activate.0(V2))), activate.1(V1), activate.0(V2))
activate.0(n__cons.1-0(X1, X2)) → cons.1-0(activate.1(X1), X2)
activate.1(n__length.1(X)) → length.1(activate.1(X))
isNatIList.0(n__zeros.) → tt.
activate.0(n__length.0(X)) → length.0(activate.0(X))
activate.1(n__and.0-1(X1, X2)) → and.0-1(activate.0(X1), X2)
U51.0-1-1(tt., V1, V2) → U52.0-1(isNat.1(activate.1(V1)), activate.1(V2))
U41.0-0-1(tt., V1, V2) → U42.0-1(isNat.0(activate.0(V1)), activate.1(V2))
U52.0-0(tt., V2) → U53.0(isNatList.0(activate.0(V2)))
U51.0-1-0(tt., V1, V2) → U52.0-0(isNat.1(activate.1(V1)), activate.0(V2))
isNatKind.0(n__s.1(V1)) → isNatKind.1(activate.1(V1))
isNatList.0(n__cons.0-0(V1, V2)) → U51.0-0-0(and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.0(activate.0(V2))), activate.0(V1), activate.0(V2))
nil. → n__nil.
isNatList.1(n__nil.) → tt.
activate.0(n__isNatIListKind.1(X)) → isNatIListKind.1(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.

ACTIVATE.0(n__and.1-0(X1, X2)) → ACTIVATE.1(X1)
ISNATILISTKIND.0(n__cons.1-1(n__0., y1)) → AND.0-0(isNatKind.1(0.), n__isNatIListKind.1(activate.1(y1)))
ISNATILISTKIND.0(n__cons.0-1(n__and.1-0(x0, x1), y1)) → AND.0-0(isNatKind.0(and.1-0(activate.1(x0), x1)), n__isNatIListKind.1(activate.1(y1)))
ACTIVATE.0(n__and.0-0(n__and.1-0(x0, x1), y1)) → AND.0-0(and.1-0(activate.1(x0), x1), y1)
ISNATILISTKIND.0(n__cons.0-0(n__and.1-0(x0, x1), y1)) → AND.0-0(isNatKind.0(and.1-0(activate.1(x0), x1)), n__isNatIListKind.0(activate.0(y1)))
ACTIVATE.1(n__and.0-1(n__and.1-0(x0, x1), y1)) → AND.0-1(and.1-0(activate.1(x0), x1), y1)
ISNATILISTKIND.0(n__cons.0-1(x0, y1)) → AND.0-0(isNatKind.0(x0), n__isNatIListKind.1(activate.1(y1)))
ISNATILISTKIND.0(n__cons.0-1(n__and.0-0(x0, x1), y1)) → AND.0-0(isNatKind.0(and.0-0(activate.0(x0), x1)), n__isNatIListKind.1(activate.1(y1)))

The remaining pairs can at least be oriented weakly.

ACTIVATE.1(n__and.0-1(n__isNatKind.1(x0), y1)) → AND.0-1(isNatKind.1(x0), y1)
ISNATKIND.0(n__s.0(n__and.1-0(x0, x1))) → ISNATKIND.0(and.1-0(activate.1(x0), x1))
ACTIVATE.0(n__and.0-0(X1, X2)) → ACTIVATE.0(X1)
ISNATKIND.0(n__s.0(n__s.0(x0))) → ISNATKIND.0(s.0(activate.0(x0)))
ISNATILISTKIND.0(n__cons.1-0(n__and.0-1(x0, x1), y1)) → AND.0-0(isNatKind.1(and.0-1(activate.0(x0), x1)), n__isNatIListKind.0(activate.0(y1)))
AND.0-1(tt., X) → ACTIVATE.1(X)
ACTIVATE.0(n__isNatIListKind.0(X)) → ISNATILISTKIND.0(X)
ISNATKIND.0(n__s.0(n__and.0-0(x0, x1))) → ISNATKIND.0(and.0-0(activate.0(x0), x1))
AND.0-0(tt., X) → ACTIVATE.0(X)
ISNATILISTKIND.0(n__cons.1-0(n__0., y1)) → AND.0-0(isNatKind.1(0.), n__isNatIListKind.0(activate.0(y1)))
ACTIVATE.0(n__and.0-0(n__isNatKind.0(x0), y1)) → AND.0-0(isNatKind.0(x0), y1)
ACTIVATE.0(n__s.1(X)) → ACTIVATE.1(X)
ACTIVATE.1(n__and.0-1(X1, X2)) → ACTIVATE.0(X1)
ISNATKIND.0(n__s.1(V1)) → ACTIVATE.1(V1)
ACTIVATE.0(n__and.0-0(n__isNatKind.1(x0), y1)) → AND.0-0(isNatKind.1(x0), y1)
ACTIVATE.0(n__s.0(X)) → ACTIVATE.0(X)
ISNATILISTKIND.0(n__cons.1-0(n__and.1-1(x0, x1), y1)) → AND.0-0(isNatKind.1(and.1-1(activate.1(x0), x1)), n__isNatIListKind.0(activate.0(y1)))
ISNATILISTKIND.0(n__cons.0-0(n__and.0-0(x0, x1), y1)) → AND.0-0(isNatKind.0(and.0-0(activate.0(x0), x1)), n__isNatIListKind.0(activate.0(y1)))
ISNATKIND.0(n__s.0(x0)) → ISNATKIND.0(x0)
ACTIVATE.0(n__and.0-0(n__and.0-0(x0, x1), y1)) → AND.0-0(and.0-0(activate.0(x0), x1), y1)
ISNATILISTKIND.0(n__cons.1-1(x0, y1)) → AND.0-0(isNatKind.1(x0), n__isNatIListKind.1(activate.1(y1)))
ACTIVATE.0(n__and.0-0(x0, y1)) → AND.0-0(x0, y1)
ISNATILISTKIND.0(n__cons.1-1(n__and.0-1(x0, x1), y1)) → AND.0-0(isNatKind.1(and.0-1(activate.0(x0), x1)), n__isNatIListKind.1(activate.1(y1)))
ISNATILISTKIND.0(n__cons.1-0(x0, y1)) → AND.0-0(isNatKind.1(x0), n__isNatIListKind.0(activate.0(y1)))
ISNATKIND.0(n__s.0(n__length.0(x0))) → ISNATKIND.0(length.0(activate.0(x0)))
ACTIVATE.1(n__and.1-1(X1, X2)) → ACTIVATE.1(X1)
ISNATILISTKIND.0(n__cons.1-1(n__and.1-1(x0, x1), y1)) → AND.0-0(isNatKind.1(and.1-1(activate.1(x0), x1)), n__isNatIListKind.1(activate.1(y1)))
ACTIVATE.0(n__isNatKind.0(X)) → ISNATKIND.0(X)
ACTIVATE.1(n__and.0-1(x0, y1)) → AND.0-1(x0, y1)
ISNATILISTKIND.0(n__cons.0-0(x0, y1)) → AND.0-0(isNatKind.0(x0), n__isNatIListKind.0(activate.0(y1)))
ACTIVATE.1(n__and.0-1(n__isNatKind.0(x0), y1)) → AND.0-1(isNatKind.0(x0), y1)
ACTIVATE.1(n__and.0-1(n__and.0-0(x0, x1), y1)) → AND.0-1(and.0-0(activate.0(x0), x1), y1)
ISNATKIND.0(n__s.0(n__s.1(x0))) → ISNATKIND.0(s.1(activate.1(x0)))
ISNATKIND.0(n__s.0(V1)) → ACTIVATE.0(V1)

Used ordering: Polynomial interpretation [25]:

POL(0.) = 1
POL(ACTIVATE.0(x₁)) = x₁
POL(ACTIVATE.1(x₁)) = x₁
POL(AND.0-0(x₁, x₂)) = x₂
POL(AND.0-1(x₁, x₂)) = x₂
POL(ISNATILISTKIND.0(x₁)) = x₁
POL(ISNATKIND.0(x₁)) = x₁
POL(U11.0-0(x₁, x₂)) = 0
POL(U11.0-1(x₁, x₂)) = 0
POL(U12.0(x₁)) = 0
POL(U21.0-0(x₁, x₂)) = 0
POL(U21.0-1(x₁, x₂)) = 0
POL(U22.0(x₁)) = 0
POL(U51.0-0-0(x₁, x₂, x₃)) = 1
POL(U51.0-0-1(x₁, x₂, x₃)) = 1
POL(U51.0-1-0(x₁, x₂, x₃)) = 1
POL(U51.0-1-1(x₁, x₂, x₃)) = 1 + x₂
POL(U52.0-0(x₁, x₂)) = 1
POL(U52.0-1(x₁, x₂)) = 1
POL(U53.0(x₁)) = 0
POL(U61.0-0(x₁, x₂)) = x₂
POL(U61.0-1(x₁, x₂)) = x₂
POL(activate.0(x₁)) = x₁
POL(activate.1(x₁)) = x₁
POL(and.0-0(x₁, x₂)) = x₁ + x₂
POL(and.0-1(x₁, x₂)) = x₁ + x₂
POL(and.1-0(x₁, x₂)) = 1 + x₁
POL(and.1-1(x₁, x₂)) = x₁ + x₂
POL(cons.0-0(x₁, x₂)) = x₁ + x₂
POL(cons.0-1(x₁, x₂)) = 1 + x₁ + x₂
POL(cons.1-0(x₁, x₂)) = x₂
POL(cons.1-1(x₁, x₂)) = x₁ + x₂
POL(isNat.0(x₁)) = 0
POL(isNat.1(x₁)) = x₁
POL(isNatIListKind.0(x₁)) = x₁
POL(isNatIListKind.1(x₁)) = 0
POL(isNatKind.0(x₁)) = x₁
POL(isNatKind.1(x₁)) = 0
POL(isNatList.0(x₁)) = 1 + x₁
POL(isNatList.1(x₁)) = 0
POL(length.0(x₁)) = x₁
POL(length.1(x₁)) = x₁
POL(n__0.) = 1
POL(n__and.0-0(x₁, x₂)) = x₁ + x₂
POL(n__and.0-1(x₁, x₂)) = x₁ + x₂
POL(n__and.1-0(x₁, x₂)) = 1 + x₁
POL(n__and.1-1(x₁, x₂)) = x₁ + x₂
POL(n__cons.0-0(x₁, x₂)) = x₁ + x₂
POL(n__cons.0-1(x₁, x₂)) = 1 + x₁ + x₂
POL(n__cons.1-0(x₁, x₂)) = x₂
POL(n__cons.1-1(x₁, x₂)) = x₁ + x₂
POL(n__isNat.0(x₁)) = 0
POL(n__isNat.1(x₁)) = x₁
POL(n__isNatIListKind.0(x₁)) = x₁
POL(n__isNatIListKind.1(x₁)) = 0
POL(n__isNatKind.0(x₁)) = x₁
POL(n__isNatKind.1(x₁)) = 0
POL(n__length.0(x₁)) = x₁
POL(n__length.1(x₁)) = x₁
POL(n__nil.) = 1
POL(n__s.0(x₁)) = x₁
POL(n__s.1(x₁)) = x₁
POL(n__zeros.) = 1
POL(nil.) = 1
POL(s.0(x₁)) = x₁
POL(s.1(x₁)) = x₁
POL(tt.) = 0
POL(zeros.) = 1

The following usable rules [17] were oriented:

length.1(X) → n__length.1(X)
cons.1-1(X1, X2) → n__cons.1-1(X1, X2)
and.1-0(X1, X2) → n__and.1-0(X1, X2)
activate.0(X) → X
U61.0-0(tt., L) → s.0(length.0(activate.0(L)))
U22.0(tt.) → tt.
isNatIListKind.1(X) → n__isNatIListKind.1(X)
U51.0-0-0(tt., V1, V2) → U52.0-0(isNat.0(activate.0(V1)), activate.0(V2))
isNatIListKind.0(n__zeros.) → tt.
activate.1(n__nil.) → nil.
activate.0(n__cons.1-1(X1, X2)) → cons.1-1(activate.1(X1), X2)
activate.0(n__cons.0-0(X1, X2)) → cons.0-0(activate.0(X1), X2)
isNatIListKind.0(X) → n__isNatIListKind.0(X)
isNat.0(n__length.0(V1)) → U11.0-0(isNatIListKind.0(activate.0(V1)), activate.0(V1))
s.1(X) → n__s.1(X)
isNat.0(n__s.1(V1)) → U21.0-1(isNatKind.1(activate.1(V1)), activate.1(V1))
zeros. → n__zeros.
U11.0-0(tt., V1) → U12.0(isNatList.0(activate.0(V1)))
U11.0-1(tt., V1) → U12.0(isNatList.1(activate.1(V1)))
activate.0(n__isNatKind.1(X)) → isNatKind.1(X)
activate.1(n__and.1-1(X1, X2)) → and.1-1(activate.1(X1), X2)
isNat.1(X) → n__isNat.1(X)
activate.0(n__isNatIListKind.1(X)) → isNatIListKind.1(X)
nil. → n__nil.
U51.0-1-1(tt., V1, V2) → U52.0-1(isNat.1(activate.1(V1)), activate.1(V2))
activate.0(n__length.0(X)) → length.0(activate.0(X))
and.0-1(tt., X) → activate.1(X)
activate.1(n__and.0-1(X1, X2)) → and.0-1(activate.0(X1), X2)
isNatKind.0(n__s.1(V1)) → isNatKind.1(activate.1(V1))
isNatList.0(n__cons.0-0(V1, V2)) → U51.0-0-0(and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.0(activate.0(V2))), activate.0(V1), activate.0(V2))
U52.0-0(tt., V2) → U53.0(isNatList.0(activate.0(V2)))
U51.0-1-0(tt., V1, V2) → U52.0-0(isNat.1(activate.1(V1)), activate.0(V2))
activate.1(n__length.1(X)) → length.1(activate.1(X))
activate.0(n__cons.1-0(X1, X2)) → cons.1-0(activate.1(X1), X2)
isNatList.0(n__cons.1-0(V1, V2)) → U51.0-1-0(and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.0(activate.0(V2))), activate.1(V1), activate.0(V2))
isNatKind.1(n__0.) → tt.
isNatIListKind.1(n__nil.) → tt.
activate.1(X) → X
activate.0(n__isNat.0(X)) → isNat.0(X)
isNatKind.1(X) → n__isNatKind.1(X)
length.0(cons.1-0(N, L)) → U61.0-0(and.0-0(and.0-0(isNatList.0(activate.0(L)), n__isNatIListKind.0(activate.0(L))), n__and.0-0(n__isNat.1(N), n__isNatKind.1(N))), activate.0(L))
0. → n__0.
isNatKind.1(n__length.1(V1)) → isNatIListKind.1(activate.1(V1))
and.0-0(X1, X2) → n__and.0-0(X1, X2)
activate.0(n__s.0(X)) → s.0(activate.0(X))
isNat.0(n__s.0(V1)) → U21.0-0(isNatKind.0(activate.0(V1)), activate.0(V1))
activate.0(n__zeros.) → zeros.
length.0(cons.0-1(N, L)) → U61.0-1(and.0-0(and.0-0(isNatList.1(activate.1(L)), n__isNatIListKind.1(activate.1(L))), n__and.0-0(n__isNat.0(N), n__isNatKind.0(N))), activate.1(L))
U53.0(tt.) → tt.
activate.0(n__and.1-0(X1, X2)) → and.1-0(activate.1(X1), X2)
activate.1(n__0.) → 0.
U12.0(tt.) → tt.
U61.0-1(tt., L) → s.1(length.1(activate.1(L)))
and.0-1(X1, X2) → n__and.0-1(X1, X2)
U21.0-0(tt., V1) → U22.0(isNat.0(activate.0(V1)))
length.0(cons.1-1(N, L)) → U61.0-1(and.0-0(and.0-0(isNatList.1(activate.1(L)), n__isNatIListKind.1(activate.1(L))), n__and.0-0(n__isNat.1(N), n__isNatKind.1(N))), activate.1(L))
isNat.1(n__0.) → tt.
activate.0(n__cons.0-1(X1, X2)) → cons.0-1(activate.0(X1), X2)
isNat.1(n__length.1(V1)) → U11.0-1(isNatIListKind.1(activate.1(V1)), activate.1(V1))
isNatKind.0(X) → n__isNatKind.0(X)
s.0(X) → n__s.0(X)
cons.0-1(X1, X2) → n__cons.0-1(X1, X2)
cons.0-0(X1, X2) → n__cons.0-0(X1, X2)
activate.0(n__isNat.1(X)) → isNat.1(X)
zeros. → cons.1-0(0., n__zeros.)
length.0(cons.0-0(N, L)) → U61.0-0(and.0-0(and.0-0(isNatList.0(activate.0(L)), n__isNatIListKind.0(activate.0(L))), n__and.0-0(n__isNat.0(N), n__isNatKind.0(N))), activate.0(L))
isNat.0(X) → n__isNat.0(X)
activate.0(n__s.1(X)) → s.1(activate.1(X))
isNatIListKind.0(n__cons.1-0(V1, V2)) → and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.0(activate.0(V2)))
isNatIListKind.0(n__cons.1-1(V1, V2)) → and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.1(activate.1(V2)))
isNatIListKind.0(n__cons.0-0(V1, V2)) → and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.0(activate.0(V2)))
isNatIListKind.0(n__cons.0-1(V1, V2)) → and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.1(activate.1(V2)))
and.0-0(tt., X) → activate.0(X)
activate.0(n__and.0-0(X1, X2)) → and.0-0(activate.0(X1), X2)
isNatKind.0(n__s.0(V1)) → isNatKind.0(activate.0(V1))
isNatKind.0(n__length.0(V1)) → isNatIListKind.0(activate.0(V1))
activate.0(n__isNatIListKind.0(X)) → isNatIListKind.0(X)
activate.0(n__isNatKind.0(X)) → isNatKind.0(X)
and.1-1(X1, X2) → n__and.1-1(X1, X2)
length.1(nil.) → 0.
cons.1-0(X1, X2) → n__cons.1-0(X1, X2)
U52.0-1(tt., V2) → U53.0(isNatList.1(activate.1(V2)))
isNatList.0(n__cons.0-1(V1, V2)) → U51.0-0-1(and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.1(activate.1(V2))), activate.0(V1), activate.1(V2))
length.0(X) → n__length.0(X)
isNatList.0(n__cons.1-1(V1, V2)) → U51.0-1-1(and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.1(activate.1(V2))), activate.1(V1), activate.1(V2))
U21.0-1(tt., V1) → U22.0(isNat.1(activate.1(V1)))
U51.0-0-1(tt., V1, V2) → U52.0-1(isNat.0(activate.0(V1)), activate.1(V2))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ 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

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ SemLabProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                                                            ↳ SemLabProof2

                              ↳ QDP

                    ↳ QDP

          ↳ QDP

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

ACTIVATE.1(n__and.0-1(n__isNatKind.1(x0), y1)) → AND.0-1(isNatKind.1(x0), y1)
ISNATKIND.0(n__s.0(n__and.1-0(x0, x1))) → ISNATKIND.0(and.1-0(activate.1(x0), x1))
ACTIVATE.0(n__and.0-0(X1, X2)) → ACTIVATE.0(X1)
ISNATKIND.0(n__s.0(n__s.0(x0))) → ISNATKIND.0(s.0(activate.0(x0)))
ISNATILISTKIND.0(n__cons.1-0(n__and.0-1(x0, x1), y1)) → AND.0-0(isNatKind.1(and.0-1(activate.0(x0), x1)), n__isNatIListKind.0(activate.0(y1)))
AND.0-1(tt., X) → ACTIVATE.1(X)
ACTIVATE.0(n__isNatIListKind.0(X)) → ISNATILISTKIND.0(X)
ISNATKIND.0(n__s.0(n__and.0-0(x0, x1))) → ISNATKIND.0(and.0-0(activate.0(x0), x1))
AND.0-0(tt., X) → ACTIVATE.0(X)
ISNATILISTKIND.0(n__cons.1-0(n__0., y1)) → AND.0-0(isNatKind.1(0.), n__isNatIListKind.0(activate.0(y1)))
ACTIVATE.0(n__and.0-0(n__isNatKind.0(x0), y1)) → AND.0-0(isNatKind.0(x0), y1)
ACTIVATE.0(n__s.1(X)) → ACTIVATE.1(X)
ACTIVATE.1(n__and.0-1(X1, X2)) → ACTIVATE.0(X1)
ISNATKIND.0(n__s.1(V1)) → ACTIVATE.1(V1)
ACTIVATE.0(n__and.0-0(n__isNatKind.1(x0), y1)) → AND.0-0(isNatKind.1(x0), y1)
ACTIVATE.0(n__s.0(X)) → ACTIVATE.0(X)
ISNATILISTKIND.0(n__cons.1-0(n__and.1-1(x0, x1), y1)) → AND.0-0(isNatKind.1(and.1-1(activate.1(x0), x1)), n__isNatIListKind.0(activate.0(y1)))
ISNATILISTKIND.0(n__cons.0-0(n__and.0-0(x0, x1), y1)) → AND.0-0(isNatKind.0(and.0-0(activate.0(x0), x1)), n__isNatIListKind.0(activate.0(y1)))
ISNATKIND.0(n__s.0(x0)) → ISNATKIND.0(x0)
ACTIVATE.0(n__and.0-0(n__and.0-0(x0, x1), y1)) → AND.0-0(and.0-0(activate.0(x0), x1), y1)
ISNATILISTKIND.0(n__cons.1-1(x0, y1)) → AND.0-0(isNatKind.1(x0), n__isNatIListKind.1(activate.1(y1)))
ACTIVATE.0(n__and.0-0(x0, y1)) → AND.0-0(x0, y1)
ISNATILISTKIND.0(n__cons.1-1(n__and.0-1(x0, x1), y1)) → AND.0-0(isNatKind.1(and.0-1(activate.0(x0), x1)), n__isNatIListKind.1(activate.1(y1)))
ISNATILISTKIND.0(n__cons.1-0(x0, y1)) → AND.0-0(isNatKind.1(x0), n__isNatIListKind.0(activate.0(y1)))
ACTIVATE.1(n__and.1-1(X1, X2)) → ACTIVATE.1(X1)
ISNATKIND.0(n__s.0(n__length.0(x0))) → ISNATKIND.0(length.0(activate.0(x0)))
ACTIVATE.0(n__isNatKind.0(X)) → ISNATKIND.0(X)
ISNATILISTKIND.0(n__cons.1-1(n__and.1-1(x0, x1), y1)) → AND.0-0(isNatKind.1(and.1-1(activate.1(x0), x1)), n__isNatIListKind.1(activate.1(y1)))
ACTIVATE.1(n__and.0-1(n__isNatKind.0(x0), y1)) → AND.0-1(isNatKind.0(x0), y1)
ISNATILISTKIND.0(n__cons.0-0(x0, y1)) → AND.0-0(isNatKind.0(x0), n__isNatIListKind.0(activate.0(y1)))
ACTIVATE.1(n__and.0-1(x0, y1)) → AND.0-1(x0, y1)
ACTIVATE.1(n__and.0-1(n__and.0-0(x0, x1), y1)) → AND.0-1(and.0-0(activate.0(x0), x1), y1)
ISNATKIND.0(n__s.0(V1)) → ACTIVATE.0(V1)
ISNATKIND.0(n__s.0(n__s.1(x0))) → ISNATKIND.0(s.1(activate.1(x0)))

The TRS R consists of the following rules:

U12.0(tt.) → tt.
U61.0-1(tt., L) → s.1(length.1(activate.1(L)))
length.0(cons.0-1(N, L)) → U61.0-1(and.0-0(and.0-0(isNatList.1(activate.1(L)), n__isNatIListKind.1(activate.1(L))), n__and.0-0(n__isNat.0(N), n__isNatKind.0(N))), activate.1(L))
U53.0(tt.) → tt.
activate.0(n__and.1-0(X1, X2)) → and.1-0(activate.1(X1), X2)
activate.1(n__0.) → 0.
isNatIList.0(n__cons.0-1(V1, V2)) → U41.0-0-1(and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.1(activate.1(V2))), activate.0(V1), activate.1(V2))
isNatIList.1(V) → U31.0-1(isNatIListKind.1(activate.1(V)), activate.1(V))
activate.0(n__zeros.) → zeros.
and.0-1(tt., X) → activate.1(X)
isNatKind.1(n__length.1(V1)) → isNatIListKind.1(activate.1(V1))
and.0-0(X1, X2) → n__and.0-0(X1, X2)
activate.0(n__s.0(X)) → s.0(activate.0(X))
isNat.0(n__s.0(V1)) → U21.0-0(isNatKind.0(activate.0(V1)), activate.0(V1))
U51.0-0-1(tt., V1, V2) → U52.0-1(isNat.0(activate.0(V1)), activate.1(V2))
U21.0-1(tt., V1) → U22.0(isNat.1(activate.1(V1)))
isNatList.0(n__cons.1-1(V1, V2)) → U51.0-1-1(and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.1(activate.1(V2))), activate.1(V1), activate.1(V2))
length.0(X) → n__length.0(X)
isNatList.0(n__cons.0-1(V1, V2)) → U51.0-0-1(and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.1(activate.1(V2))), activate.0(V1), activate.1(V2))
U52.0-1(tt., V2) → U53.0(isNatList.1(activate.1(V2)))
cons.1-0(X1, X2) → n__cons.1-0(X1, X2)
length.1(nil.) → 0.
isNatIList.0(n__cons.1-1(V1, V2)) → U41.0-1-1(and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.1(activate.1(V2))), activate.1(V1), activate.1(V2))
and.1-1(X1, X2) → n__and.1-1(X1, X2)
activate.0(n__isNatKind.0(X)) → isNatKind.0(X)
activate.0(n__s.1(X)) → s.1(activate.1(X))
U31.0-0(tt., V) → U32.0(isNatList.0(activate.0(V)))
isNatIListKind.0(n__cons.0-0(V1, V2)) → and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.0(activate.0(V2)))
isNat.0(X) → n__isNat.0(X)
length.0(cons.0-0(N, L)) → U61.0-0(and.0-0(and.0-0(isNatList.0(activate.0(L)), n__isNatIListKind.0(activate.0(L))), n__and.0-0(n__isNat.0(N), n__isNatKind.0(N))), activate.0(L))
activate.0(n__and.0-0(X1, X2)) → and.0-0(activate.0(X1), X2)
zeros. → cons.1-0(0., n__zeros.)
activate.0(n__isNat.1(X)) → isNat.1(X)
activate.0(n__isNatIListKind.0(X)) → isNatIListKind.0(X)
cons.0-0(X1, X2) → n__cons.0-0(X1, X2)
isNatIListKind.0(n__cons.1-1(V1, V2)) → and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.1(activate.1(V2)))
s.0(X) → n__s.0(X)
cons.0-1(X1, X2) → n__cons.0-1(X1, X2)
U41.0-1-0(tt., V1, V2) → U42.0-0(isNat.1(activate.1(V1)), activate.0(V2))
isNatKind.0(X) → n__isNatKind.0(X)
activate.0(n__cons.0-1(X1, X2)) → cons.0-1(activate.0(X1), X2)
isNat.1(n__length.1(V1)) → U11.0-1(isNatIListKind.1(activate.1(V1)), activate.1(V1))
length.0(cons.1-1(N, L)) → U61.0-1(and.0-0(and.0-0(isNatList.1(activate.1(L)), n__isNatIListKind.1(activate.1(L))), n__and.0-0(n__isNat.1(N), n__isNatKind.1(N))), activate.1(L))
isNat.1(n__0.) → tt.
and.0-1(X1, X2) → n__and.0-1(X1, X2)
U21.0-0(tt., V1) → U22.0(isNat.0(activate.0(V1)))
isNatKind.0(n__length.0(V1)) → isNatIListKind.0(activate.0(V1))
zeros. → n__zeros.
U11.0-1(tt., V1) → U12.0(isNatList.1(activate.1(V1)))
U11.0-0(tt., V1) → U12.0(isNatList.0(activate.0(V1)))
U43.0(tt.) → tt.
activate.0(n__isNatKind.1(X)) → isNatKind.1(X)
isNat.1(X) → n__isNat.1(X)
activate.1(n__and.1-1(X1, X2)) → and.1-1(activate.1(X1), X2)
U31.0-1(tt., V) → U32.0(isNatList.1(activate.1(V)))
activate.0(n__cons.0-0(X1, X2)) → cons.0-0(activate.0(X1), X2)
U41.0-0-0(tt., V1, V2) → U42.0-0(isNat.0(activate.0(V1)), activate.0(V2))
U32.0(tt.) → tt.
isNat.0(n__length.0(V1)) → U11.0-0(isNatIListKind.0(activate.0(V1)), activate.0(V1))
isNatIListKind.0(X) → n__isNatIListKind.0(X)
isNat.0(n__s.1(V1)) → U21.0-1(isNatKind.1(activate.1(V1)), activate.1(V1))
s.1(X) → n__s.1(X)
U22.0(tt.) → tt.
isNatIListKind.1(X) → n__isNatIListKind.1(X)
U51.0-0-0(tt., V1, V2) → U52.0-0(isNat.0(activate.0(V1)), activate.0(V2))
isNatIListKind.0(n__zeros.) → tt.
activate.1(n__nil.) → nil.
isNatKind.0(n__s.0(V1)) → isNatKind.0(activate.0(V1))
isNatIList.0(n__cons.0-0(V1, V2)) → U41.0-0-0(and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.0(activate.0(V2))), activate.0(V1), activate.0(V2))
activate.0(n__cons.1-1(X1, X2)) → cons.1-1(activate.1(X1), X2)
isNatIList.0(n__cons.1-0(V1, V2)) → U41.0-1-0(and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.0(activate.0(V2))), activate.1(V1), activate.0(V2))
and.0-0(tt., X) → activate.0(X)
U42.0-1(tt., V2) → U43.0(isNatIList.1(activate.1(V2)))
length.1(X) → n__length.1(X)
cons.1-1(X1, X2) → n__cons.1-1(X1, X2)
and.1-0(X1, X2) → n__and.1-0(X1, X2)
activate.0(X) → X
U61.0-0(tt., L) → s.0(length.0(activate.0(L)))
activate.0(n__isNat.0(X)) → isNat.0(X)
activate.1(X) → X
isNatIListKind.1(n__nil.) → tt.
isNatKind.1(n__0.) → tt.
0. → n__0.
U41.0-1-1(tt., V1, V2) → U42.0-1(isNat.1(activate.1(V1)), activate.1(V2))
length.0(cons.1-0(N, L)) → U61.0-0(and.0-0(and.0-0(isNatList.0(activate.0(L)), n__isNatIListKind.0(activate.0(L))), n__and.0-0(n__isNat.1(N), n__isNatKind.1(N))), activate.0(L))
isNatKind.1(X) → n__isNatKind.1(X)
isNatIListKind.0(n__cons.1-0(V1, V2)) → and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.0(activate.0(V2)))
isNatIListKind.0(n__cons.0-1(V1, V2)) → and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.1(activate.1(V2)))
isNatIList.0(V) → U31.0-0(isNatIListKind.0(activate.0(V)), activate.0(V))
U42.0-0(tt., V2) → U43.0(isNatIList.0(activate.0(V2)))
isNatList.0(n__cons.1-0(V1, V2)) → U51.0-1-0(and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.0(activate.0(V2))), activate.1(V1), activate.0(V2))
activate.0(n__cons.1-0(X1, X2)) → cons.1-0(activate.1(X1), X2)
activate.1(n__length.1(X)) → length.1(activate.1(X))
isNatIList.0(n__zeros.) → tt.
activate.0(n__length.0(X)) → length.0(activate.0(X))
activate.1(n__and.0-1(X1, X2)) → and.0-1(activate.0(X1), X2)
U51.0-1-1(tt., V1, V2) → U52.0-1(isNat.1(activate.1(V1)), activate.1(V2))
U41.0-0-1(tt., V1, V2) → U42.0-1(isNat.0(activate.0(V1)), activate.1(V2))
U52.0-0(tt., V2) → U53.0(isNatList.0(activate.0(V2)))
U51.0-1-0(tt., V1, V2) → U52.0-0(isNat.1(activate.1(V1)), activate.0(V2))
isNatKind.0(n__s.1(V1)) → isNatKind.1(activate.1(V1))
isNatList.0(n__cons.0-0(V1, V2)) → U51.0-0-0(and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.0(activate.0(V2))), activate.0(V1), activate.0(V2))
nil. → n__nil.
isNatList.1(n__nil.) → tt.
activate.0(n__isNatIListKind.1(X)) → isNatIListKind.1(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.

ACTIVATE.1(n__and.0-1(n__isNatKind.1(x0), y1)) → AND.0-1(isNatKind.1(x0), y1)
ACTIVATE.0(n__and.0-0(n__isNatKind.0(x0), y1)) → AND.0-0(isNatKind.0(x0), y1)
ACTIVATE.1(n__and.0-1(X1, X2)) → ACTIVATE.0(X1)
ISNATILISTKIND.0(n__cons.0-0(n__and.0-0(x0, x1), y1)) → AND.0-0(isNatKind.0(and.0-0(activate.0(x0), x1)), n__isNatIListKind.0(activate.0(y1)))
ISNATILISTKIND.0(n__cons.1-1(x0, y1)) → AND.0-0(isNatKind.1(x0), n__isNatIListKind.1(activate.1(y1)))
ISNATILISTKIND.0(n__cons.1-1(n__and.0-1(x0, x1), y1)) → AND.0-0(isNatKind.1(and.0-1(activate.0(x0), x1)), n__isNatIListKind.1(activate.1(y1)))
ACTIVATE.1(n__and.1-1(X1, X2)) → ACTIVATE.1(X1)
ACTIVATE.0(n__isNatKind.0(X)) → ISNATKIND.0(X)
ISNATILISTKIND.0(n__cons.1-1(n__and.1-1(x0, x1), y1)) → AND.0-0(isNatKind.1(and.1-1(activate.1(x0), x1)), n__isNatIListKind.1(activate.1(y1)))
ACTIVATE.1(n__and.0-1(n__isNatKind.0(x0), y1)) → AND.0-1(isNatKind.0(x0), y1)
ISNATILISTKIND.0(n__cons.0-0(x0, y1)) → AND.0-0(isNatKind.0(x0), n__isNatIListKind.0(activate.0(y1)))
ACTIVATE.1(n__and.0-1(x0, y1)) → AND.0-1(x0, y1)
ACTIVATE.1(n__and.0-1(n__and.0-0(x0, x1), y1)) → AND.0-1(and.0-0(activate.0(x0), x1), y1)

The remaining pairs can at least be oriented weakly.

ISNATKIND.0(n__s.0(n__and.1-0(x0, x1))) → ISNATKIND.0(and.1-0(activate.1(x0), x1))
ACTIVATE.0(n__and.0-0(X1, X2)) → ACTIVATE.0(X1)
ISNATKIND.0(n__s.0(n__s.0(x0))) → ISNATKIND.0(s.0(activate.0(x0)))
ISNATILISTKIND.0(n__cons.1-0(n__and.0-1(x0, x1), y1)) → AND.0-0(isNatKind.1(and.0-1(activate.0(x0), x1)), n__isNatIListKind.0(activate.0(y1)))
AND.0-1(tt., X) → ACTIVATE.1(X)
ACTIVATE.0(n__isNatIListKind.0(X)) → ISNATILISTKIND.0(X)
ISNATKIND.0(n__s.0(n__and.0-0(x0, x1))) → ISNATKIND.0(and.0-0(activate.0(x0), x1))
AND.0-0(tt., X) → ACTIVATE.0(X)
ISNATILISTKIND.0(n__cons.1-0(n__0., y1)) → AND.0-0(isNatKind.1(0.), n__isNatIListKind.0(activate.0(y1)))
ACTIVATE.0(n__s.1(X)) → ACTIVATE.1(X)
ISNATKIND.0(n__s.1(V1)) → ACTIVATE.1(V1)
ACTIVATE.0(n__and.0-0(n__isNatKind.1(x0), y1)) → AND.0-0(isNatKind.1(x0), y1)
ACTIVATE.0(n__s.0(X)) → ACTIVATE.0(X)
ISNATILISTKIND.0(n__cons.1-0(n__and.1-1(x0, x1), y1)) → AND.0-0(isNatKind.1(and.1-1(activate.1(x0), x1)), n__isNatIListKind.0(activate.0(y1)))
ISNATKIND.0(n__s.0(x0)) → ISNATKIND.0(x0)
ACTIVATE.0(n__and.0-0(n__and.0-0(x0, x1), y1)) → AND.0-0(and.0-0(activate.0(x0), x1), y1)
ACTIVATE.0(n__and.0-0(x0, y1)) → AND.0-0(x0, y1)
ISNATILISTKIND.0(n__cons.1-0(x0, y1)) → AND.0-0(isNatKind.1(x0), n__isNatIListKind.0(activate.0(y1)))
ISNATKIND.0(n__s.0(n__length.0(x0))) → ISNATKIND.0(length.0(activate.0(x0)))
ISNATKIND.0(n__s.0(V1)) → ACTIVATE.0(V1)
ISNATKIND.0(n__s.0(n__s.1(x0))) → ISNATKIND.0(s.1(activate.1(x0)))

Used ordering: Polynomial interpretation [25]:

POL(0.) = 0
POL(ACTIVATE.0(x₁)) = x₁
POL(ACTIVATE.1(x₁)) = x₁
POL(AND.0-0(x₁, x₂)) = x₂
POL(AND.0-1(x₁, x₂)) = x₂
POL(ISNATILISTKIND.0(x₁)) = x₁
POL(ISNATKIND.0(x₁)) = x₁
POL(U11.0-0(x₁, x₂)) = 0
POL(U11.0-1(x₁, x₂)) = 0
POL(U12.0(x₁)) = 0
POL(U21.0-0(x₁, x₂)) = 0
POL(U21.0-1(x₁, x₂)) = 0
POL(U22.0(x₁)) = 0
POL(U51.0-0-0(x₁, x₂, x₃)) = x₃
POL(U51.0-0-1(x₁, x₂, x₃)) = x₃
POL(U51.0-1-0(x₁, x₂, x₃)) = 0
POL(U51.0-1-1(x₁, x₂, x₃)) = 0
POL(U52.0-0(x₁, x₂)) = 0
POL(U52.0-1(x₁, x₂)) = 0
POL(U53.0(x₁)) = 0
POL(U61.0-0(x₁, x₂)) = x₂
POL(U61.0-1(x₁, x₂)) = 0
POL(activate.0(x₁)) = x₁
POL(activate.1(x₁)) = x₁
POL(and.0-0(x₁, x₂)) = x₁ + x₂
POL(and.0-1(x₁, x₂)) = 1 + x₁ + x₂
POL(and.1-0(x₁, x₂)) = 0
POL(and.1-1(x₁, x₂)) = 1 + x₁ + x₂
POL(cons.0-0(x₁, x₂)) = 1 + x₁ + x₂
POL(cons.0-1(x₁, x₂)) = 1 + x₁ + x₂
POL(cons.1-0(x₁, x₂)) = x₂
POL(cons.1-1(x₁, x₂)) = 1 + x₁ + x₂
POL(isNat.0(x₁)) = 0
POL(isNat.1(x₁)) = 0
POL(isNatIListKind.0(x₁)) = x₁
POL(isNatIListKind.1(x₁)) = 0
POL(isNatKind.0(x₁)) = 1 + x₁
POL(isNatKind.1(x₁)) = 0
POL(isNatList.0(x₁)) = x₁
POL(isNatList.1(x₁)) = 0
POL(length.0(x₁)) = x₁
POL(length.1(x₁)) = 0
POL(n__0.) = 0
POL(n__and.0-0(x₁, x₂)) = x₁ + x₂
POL(n__and.0-1(x₁, x₂)) = 1 + x₁ + x₂
POL(n__and.1-0(x₁, x₂)) = 0
POL(n__and.1-1(x₁, x₂)) = 1 + x₁ + x₂
POL(n__cons.0-0(x₁, x₂)) = 1 + x₁ + x₂
POL(n__cons.0-1(x₁, x₂)) = 1 + x₁ + x₂
POL(n__cons.1-0(x₁, x₂)) = x₂
POL(n__cons.1-1(x₁, x₂)) = 1 + x₁ + x₂
POL(n__isNat.0(x₁)) = 0
POL(n__isNat.1(x₁)) = 0
POL(n__isNatIListKind.0(x₁)) = x₁
POL(n__isNatIListKind.1(x₁)) = 0
POL(n__isNatKind.0(x₁)) = 1 + x₁
POL(n__isNatKind.1(x₁)) = 0
POL(n__length.0(x₁)) = x₁
POL(n__length.1(x₁)) = 0
POL(n__nil.) = 1
POL(n__s.0(x₁)) = x₁
POL(n__s.1(x₁)) = x₁
POL(n__zeros.) = 0
POL(nil.) = 1
POL(s.0(x₁)) = x₁
POL(s.1(x₁)) = x₁
POL(tt.) = 0
POL(zeros.) = 0

The following usable rules [17] were oriented:

length.1(X) → n__length.1(X)
cons.1-1(X1, X2) → n__cons.1-1(X1, X2)
and.1-0(X1, X2) → n__and.1-0(X1, X2)
activate.0(X) → X
U61.0-0(tt., L) → s.0(length.0(activate.0(L)))
U22.0(tt.) → tt.
isNatIListKind.1(X) → n__isNatIListKind.1(X)
U51.0-0-0(tt., V1, V2) → U52.0-0(isNat.0(activate.0(V1)), activate.0(V2))
isNatIListKind.0(n__zeros.) → tt.
activate.1(n__nil.) → nil.
activate.0(n__cons.1-1(X1, X2)) → cons.1-1(activate.1(X1), X2)
activate.0(n__cons.0-0(X1, X2)) → cons.0-0(activate.0(X1), X2)
isNatIListKind.0(X) → n__isNatIListKind.0(X)
isNat.0(n__length.0(V1)) → U11.0-0(isNatIListKind.0(activate.0(V1)), activate.0(V1))
s.1(X) → n__s.1(X)
isNat.0(n__s.1(V1)) → U21.0-1(isNatKind.1(activate.1(V1)), activate.1(V1))
zeros. → n__zeros.
U11.0-0(tt., V1) → U12.0(isNatList.0(activate.0(V1)))
U11.0-1(tt., V1) → U12.0(isNatList.1(activate.1(V1)))
activate.0(n__isNatKind.1(X)) → isNatKind.1(X)
activate.1(n__and.1-1(X1, X2)) → and.1-1(activate.1(X1), X2)
isNat.1(X) → n__isNat.1(X)
activate.0(n__isNatIListKind.1(X)) → isNatIListKind.1(X)
nil. → n__nil.
U51.0-1-1(tt., V1, V2) → U52.0-1(isNat.1(activate.1(V1)), activate.1(V2))
activate.0(n__length.0(X)) → length.0(activate.0(X))
and.0-1(tt., X) → activate.1(X)
activate.1(n__and.0-1(X1, X2)) → and.0-1(activate.0(X1), X2)
isNatKind.0(n__s.1(V1)) → isNatKind.1(activate.1(V1))
isNatList.0(n__cons.0-0(V1, V2)) → U51.0-0-0(and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.0(activate.0(V2))), activate.0(V1), activate.0(V2))
U52.0-0(tt., V2) → U53.0(isNatList.0(activate.0(V2)))
U51.0-1-0(tt., V1, V2) → U52.0-0(isNat.1(activate.1(V1)), activate.0(V2))
activate.1(n__length.1(X)) → length.1(activate.1(X))
activate.0(n__cons.1-0(X1, X2)) → cons.1-0(activate.1(X1), X2)
isNatList.0(n__cons.1-0(V1, V2)) → U51.0-1-0(and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.0(activate.0(V2))), activate.1(V1), activate.0(V2))
isNatKind.1(n__0.) → tt.
isNatIListKind.1(n__nil.) → tt.
activate.1(X) → X
activate.0(n__isNat.0(X)) → isNat.0(X)
isNatKind.1(X) → n__isNatKind.1(X)
length.0(cons.1-0(N, L)) → U61.0-0(and.0-0(and.0-0(isNatList.0(activate.0(L)), n__isNatIListKind.0(activate.0(L))), n__and.0-0(n__isNat.1(N), n__isNatKind.1(N))), activate.0(L))
0. → n__0.
isNatKind.1(n__length.1(V1)) → isNatIListKind.1(activate.1(V1))
and.0-0(X1, X2) → n__and.0-0(X1, X2)
activate.0(n__s.0(X)) → s.0(activate.0(X))
isNat.0(n__s.0(V1)) → U21.0-0(isNatKind.0(activate.0(V1)), activate.0(V1))
activate.0(n__zeros.) → zeros.
length.0(cons.0-1(N, L)) → U61.0-1(and.0-0(and.0-0(isNatList.1(activate.1(L)), n__isNatIListKind.1(activate.1(L))), n__and.0-0(n__isNat.0(N), n__isNatKind.0(N))), activate.1(L))
U53.0(tt.) → tt.
activate.0(n__and.1-0(X1, X2)) → and.1-0(activate.1(X1), X2)
activate.1(n__0.) → 0.
U12.0(tt.) → tt.
U61.0-1(tt., L) → s.1(length.1(activate.1(L)))
and.0-1(X1, X2) → n__and.0-1(X1, X2)
U21.0-0(tt., V1) → U22.0(isNat.0(activate.0(V1)))
length.0(cons.1-1(N, L)) → U61.0-1(and.0-0(and.0-0(isNatList.1(activate.1(L)), n__isNatIListKind.1(activate.1(L))), n__and.0-0(n__isNat.1(N), n__isNatKind.1(N))), activate.1(L))
isNat.1(n__0.) → tt.
activate.0(n__cons.0-1(X1, X2)) → cons.0-1(activate.0(X1), X2)
isNat.1(n__length.1(V1)) → U11.0-1(isNatIListKind.1(activate.1(V1)), activate.1(V1))
isNatKind.0(X) → n__isNatKind.0(X)
s.0(X) → n__s.0(X)
cons.0-1(X1, X2) → n__cons.0-1(X1, X2)
cons.0-0(X1, X2) → n__cons.0-0(X1, X2)
activate.0(n__isNat.1(X)) → isNat.1(X)
zeros. → cons.1-0(0., n__zeros.)
length.0(cons.0-0(N, L)) → U61.0-0(and.0-0(and.0-0(isNatList.0(activate.0(L)), n__isNatIListKind.0(activate.0(L))), n__and.0-0(n__isNat.0(N), n__isNatKind.0(N))), activate.0(L))
isNat.0(X) → n__isNat.0(X)
activate.0(n__s.1(X)) → s.1(activate.1(X))
isNatIListKind.0(n__cons.1-0(V1, V2)) → and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.0(activate.0(V2)))
isNatIListKind.0(n__cons.1-1(V1, V2)) → and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.1(activate.1(V2)))
isNatIListKind.0(n__cons.0-0(V1, V2)) → and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.0(activate.0(V2)))
isNatIListKind.0(n__cons.0-1(V1, V2)) → and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.1(activate.1(V2)))
and.0-0(tt., X) → activate.0(X)
activate.0(n__and.0-0(X1, X2)) → and.0-0(activate.0(X1), X2)
isNatKind.0(n__s.0(V1)) → isNatKind.0(activate.0(V1))
isNatKind.0(n__length.0(V1)) → isNatIListKind.0(activate.0(V1))
activate.0(n__isNatIListKind.0(X)) → isNatIListKind.0(X)
activate.0(n__isNatKind.0(X)) → isNatKind.0(X)
and.1-1(X1, X2) → n__and.1-1(X1, X2)
length.1(nil.) → 0.
cons.1-0(X1, X2) → n__cons.1-0(X1, X2)
U52.0-1(tt., V2) → U53.0(isNatList.1(activate.1(V2)))
isNatList.0(n__cons.0-1(V1, V2)) → U51.0-0-1(and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.1(activate.1(V2))), activate.0(V1), activate.1(V2))
length.0(X) → n__length.0(X)
isNatList.0(n__cons.1-1(V1, V2)) → U51.0-1-1(and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.1(activate.1(V2))), activate.1(V1), activate.1(V2))
U21.0-1(tt., V1) → U22.0(isNat.1(activate.1(V1)))
U51.0-0-1(tt., V1, V2) → U52.0-1(isNat.0(activate.0(V1)), activate.1(V2))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ 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

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ SemLabProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                            ↳ SemLabProof2

                              ↳ QDP

                    ↳ QDP

          ↳ QDP

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

ISNATKIND.0(n__s.0(n__and.1-0(x0, x1))) → ISNATKIND.0(and.1-0(activate.1(x0), x1))
ACTIVATE.0(n__and.0-0(X1, X2)) → ACTIVATE.0(X1)
ISNATKIND.0(n__s.0(n__s.0(x0))) → ISNATKIND.0(s.0(activate.0(x0)))
ISNATKIND.0(n__s.1(V1)) → ACTIVATE.1(V1)
ACTIVATE.0(n__and.0-0(n__isNatKind.1(x0), y1)) → AND.0-0(isNatKind.1(x0), y1)
ISNATILISTKIND.0(n__cons.1-0(n__and.1-1(x0, x1), y1)) → AND.0-0(isNatKind.1(and.1-1(activate.1(x0), x1)), n__isNatIListKind.0(activate.0(y1)))
ACTIVATE.0(n__s.0(X)) → ACTIVATE.0(X)
ISNATILISTKIND.0(n__cons.1-0(n__and.0-1(x0, x1), y1)) → AND.0-0(isNatKind.1(and.0-1(activate.0(x0), x1)), n__isNatIListKind.0(activate.0(y1)))
ISNATKIND.0(n__s.0(x0)) → ISNATKIND.0(x0)
ACTIVATE.0(n__and.0-0(n__and.0-0(x0, x1), y1)) → AND.0-0(and.0-0(activate.0(x0), x1), y1)
AND.0-1(tt., X) → ACTIVATE.1(X)
ACTIVATE.0(n__isNatIListKind.0(X)) → ISNATILISTKIND.0(X)
ISNATKIND.0(n__s.0(n__and.0-0(x0, x1))) → ISNATKIND.0(and.0-0(activate.0(x0), x1))
ACTIVATE.0(n__and.0-0(x0, y1)) → AND.0-0(x0, y1)
AND.0-0(tt., X) → ACTIVATE.0(X)
ISNATILISTKIND.0(n__cons.1-0(x0, y1)) → AND.0-0(isNatKind.1(x0), n__isNatIListKind.0(activate.0(y1)))
ISNATKIND.0(n__s.0(n__length.0(x0))) → ISNATKIND.0(length.0(activate.0(x0)))
ISNATILISTKIND.0(n__cons.1-0(n__0., y1)) → AND.0-0(isNatKind.1(0.), n__isNatIListKind.0(activate.0(y1)))
ACTIVATE.0(n__s.1(X)) → ACTIVATE.1(X)
ISNATKIND.0(n__s.0(n__s.1(x0))) → ISNATKIND.0(s.1(activate.1(x0)))
ISNATKIND.0(n__s.0(V1)) → ACTIVATE.0(V1)

The TRS R consists of the following rules:

U12.0(tt.) → tt.
U61.0-1(tt., L) → s.1(length.1(activate.1(L)))
length.0(cons.0-1(N, L)) → U61.0-1(and.0-0(and.0-0(isNatList.1(activate.1(L)), n__isNatIListKind.1(activate.1(L))), n__and.0-0(n__isNat.0(N), n__isNatKind.0(N))), activate.1(L))
U53.0(tt.) → tt.
activate.0(n__and.1-0(X1, X2)) → and.1-0(activate.1(X1), X2)
activate.1(n__0.) → 0.
isNatIList.0(n__cons.0-1(V1, V2)) → U41.0-0-1(and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.1(activate.1(V2))), activate.0(V1), activate.1(V2))
isNatIList.1(V) → U31.0-1(isNatIListKind.1(activate.1(V)), activate.1(V))
activate.0(n__zeros.) → zeros.
and.0-1(tt., X) → activate.1(X)
isNatKind.1(n__length.1(V1)) → isNatIListKind.1(activate.1(V1))
and.0-0(X1, X2) → n__and.0-0(X1, X2)
activate.0(n__s.0(X)) → s.0(activate.0(X))
isNat.0(n__s.0(V1)) → U21.0-0(isNatKind.0(activate.0(V1)), activate.0(V1))
U51.0-0-1(tt., V1, V2) → U52.0-1(isNat.0(activate.0(V1)), activate.1(V2))
U21.0-1(tt., V1) → U22.0(isNat.1(activate.1(V1)))
isNatList.0(n__cons.1-1(V1, V2)) → U51.0-1-1(and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.1(activate.1(V2))), activate.1(V1), activate.1(V2))
length.0(X) → n__length.0(X)
isNatList.0(n__cons.0-1(V1, V2)) → U51.0-0-1(and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.1(activate.1(V2))), activate.0(V1), activate.1(V2))
U52.0-1(tt., V2) → U53.0(isNatList.1(activate.1(V2)))
cons.1-0(X1, X2) → n__cons.1-0(X1, X2)
length.1(nil.) → 0.
isNatIList.0(n__cons.1-1(V1, V2)) → U41.0-1-1(and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.1(activate.1(V2))), activate.1(V1), activate.1(V2))
and.1-1(X1, X2) → n__and.1-1(X1, X2)
activate.0(n__isNatKind.0(X)) → isNatKind.0(X)
activate.0(n__s.1(X)) → s.1(activate.1(X))
U31.0-0(tt., V) → U32.0(isNatList.0(activate.0(V)))
isNatIListKind.0(n__cons.0-0(V1, V2)) → and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.0(activate.0(V2)))
isNat.0(X) → n__isNat.0(X)
length.0(cons.0-0(N, L)) → U61.0-0(and.0-0(and.0-0(isNatList.0(activate.0(L)), n__isNatIListKind.0(activate.0(L))), n__and.0-0(n__isNat.0(N), n__isNatKind.0(N))), activate.0(L))
activate.0(n__and.0-0(X1, X2)) → and.0-0(activate.0(X1), X2)
zeros. → cons.1-0(0., n__zeros.)
activate.0(n__isNat.1(X)) → isNat.1(X)
activate.0(n__isNatIListKind.0(X)) → isNatIListKind.0(X)
cons.0-0(X1, X2) → n__cons.0-0(X1, X2)
isNatIListKind.0(n__cons.1-1(V1, V2)) → and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.1(activate.1(V2)))
s.0(X) → n__s.0(X)
cons.0-1(X1, X2) → n__cons.0-1(X1, X2)
U41.0-1-0(tt., V1, V2) → U42.0-0(isNat.1(activate.1(V1)), activate.0(V2))
isNatKind.0(X) → n__isNatKind.0(X)
activate.0(n__cons.0-1(X1, X2)) → cons.0-1(activate.0(X1), X2)
isNat.1(n__length.1(V1)) → U11.0-1(isNatIListKind.1(activate.1(V1)), activate.1(V1))
length.0(cons.1-1(N, L)) → U61.0-1(and.0-0(and.0-0(isNatList.1(activate.1(L)), n__isNatIListKind.1(activate.1(L))), n__and.0-0(n__isNat.1(N), n__isNatKind.1(N))), activate.1(L))
isNat.1(n__0.) → tt.
and.0-1(X1, X2) → n__and.0-1(X1, X2)
U21.0-0(tt., V1) → U22.0(isNat.0(activate.0(V1)))
isNatKind.0(n__length.0(V1)) → isNatIListKind.0(activate.0(V1))
zeros. → n__zeros.
U11.0-1(tt., V1) → U12.0(isNatList.1(activate.1(V1)))
U11.0-0(tt., V1) → U12.0(isNatList.0(activate.0(V1)))
U43.0(tt.) → tt.
activate.0(n__isNatKind.1(X)) → isNatKind.1(X)
isNat.1(X) → n__isNat.1(X)
activate.1(n__and.1-1(X1, X2)) → and.1-1(activate.1(X1), X2)
U31.0-1(tt., V) → U32.0(isNatList.1(activate.1(V)))
activate.0(n__cons.0-0(X1, X2)) → cons.0-0(activate.0(X1), X2)
U41.0-0-0(tt., V1, V2) → U42.0-0(isNat.0(activate.0(V1)), activate.0(V2))
U32.0(tt.) → tt.
isNat.0(n__length.0(V1)) → U11.0-0(isNatIListKind.0(activate.0(V1)), activate.0(V1))
isNatIListKind.0(X) → n__isNatIListKind.0(X)
isNat.0(n__s.1(V1)) → U21.0-1(isNatKind.1(activate.1(V1)), activate.1(V1))
s.1(X) → n__s.1(X)
U22.0(tt.) → tt.
isNatIListKind.1(X) → n__isNatIListKind.1(X)
U51.0-0-0(tt., V1, V2) → U52.0-0(isNat.0(activate.0(V1)), activate.0(V2))
isNatIListKind.0(n__zeros.) → tt.
activate.1(n__nil.) → nil.
isNatKind.0(n__s.0(V1)) → isNatKind.0(activate.0(V1))
isNatIList.0(n__cons.0-0(V1, V2)) → U41.0-0-0(and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.0(activate.0(V2))), activate.0(V1), activate.0(V2))
activate.0(n__cons.1-1(X1, X2)) → cons.1-1(activate.1(X1), X2)
isNatIList.0(n__cons.1-0(V1, V2)) → U41.0-1-0(and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.0(activate.0(V2))), activate.1(V1), activate.0(V2))
and.0-0(tt., X) → activate.0(X)
U42.0-1(tt., V2) → U43.0(isNatIList.1(activate.1(V2)))
length.1(X) → n__length.1(X)
cons.1-1(X1, X2) → n__cons.1-1(X1, X2)
and.1-0(X1, X2) → n__and.1-0(X1, X2)
activate.0(X) → X
U61.0-0(tt., L) → s.0(length.0(activate.0(L)))
activate.0(n__isNat.0(X)) → isNat.0(X)
activate.1(X) → X
isNatIListKind.1(n__nil.) → tt.
isNatKind.1(n__0.) → tt.
0. → n__0.
U41.0-1-1(tt., V1, V2) → U42.0-1(isNat.1(activate.1(V1)), activate.1(V2))
length.0(cons.1-0(N, L)) → U61.0-0(and.0-0(and.0-0(isNatList.0(activate.0(L)), n__isNatIListKind.0(activate.0(L))), n__and.0-0(n__isNat.1(N), n__isNatKind.1(N))), activate.0(L))
isNatKind.1(X) → n__isNatKind.1(X)
isNatIListKind.0(n__cons.1-0(V1, V2)) → and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.0(activate.0(V2)))
isNatIListKind.0(n__cons.0-1(V1, V2)) → and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.1(activate.1(V2)))
isNatIList.0(V) → U31.0-0(isNatIListKind.0(activate.0(V)), activate.0(V))
U42.0-0(tt., V2) → U43.0(isNatIList.0(activate.0(V2)))
isNatList.0(n__cons.1-0(V1, V2)) → U51.0-1-0(and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.0(activate.0(V2))), activate.1(V1), activate.0(V2))
activate.0(n__cons.1-0(X1, X2)) → cons.1-0(activate.1(X1), X2)
activate.1(n__length.1(X)) → length.1(activate.1(X))
isNatIList.0(n__zeros.) → tt.
activate.0(n__length.0(X)) → length.0(activate.0(X))
activate.1(n__and.0-1(X1, X2)) → and.0-1(activate.0(X1), X2)
U51.0-1-1(tt., V1, V2) → U52.0-1(isNat.1(activate.1(V1)), activate.1(V2))
U41.0-0-1(tt., V1, V2) → U42.0-1(isNat.0(activate.0(V1)), activate.1(V2))
U52.0-0(tt., V2) → U53.0(isNatList.0(activate.0(V2)))
U51.0-1-0(tt., V1, V2) → U52.0-0(isNat.1(activate.1(V1)), activate.0(V2))
isNatKind.0(n__s.1(V1)) → isNatKind.1(activate.1(V1))
isNatList.0(n__cons.0-0(V1, V2)) → U51.0-0-0(and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.0(activate.0(V2))), activate.0(V1), activate.0(V2))
nil. → n__nil.
isNatList.1(n__nil.) → tt.
activate.0(n__isNatIListKind.1(X)) → isNatIListKind.1(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 4 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ 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

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ SemLabProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                              ↳ AND

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                            ↳ SemLabProof2

                              ↳ QDP

                    ↳ QDP

          ↳ QDP

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

ACTIVATE.0(n__and.0-0(x0, y1)) → AND.0-0(x0, y1)
ACTIVATE.0(n__and.0-0(X1, X2)) → ACTIVATE.0(X1)
AND.0-0(tt., X) → ACTIVATE.0(X)
ACTIVATE.0(n__and.0-0(n__isNatKind.1(x0), y1)) → AND.0-0(isNatKind.1(x0), y1)
ISNATILISTKIND.0(n__cons.1-0(x0, y1)) → AND.0-0(isNatKind.1(x0), n__isNatIListKind.0(activate.0(y1)))
ISNATILISTKIND.0(n__cons.1-0(n__and.1-1(x0, x1), y1)) → AND.0-0(isNatKind.1(and.1-1(activate.1(x0), x1)), n__isNatIListKind.0(activate.0(y1)))
ACTIVATE.0(n__s.0(X)) → ACTIVATE.0(X)
ISNATILISTKIND.0(n__cons.1-0(n__and.0-1(x0, x1), y1)) → AND.0-0(isNatKind.1(and.0-1(activate.0(x0), x1)), n__isNatIListKind.0(activate.0(y1)))
ISNATILISTKIND.0(n__cons.1-0(n__0., y1)) → AND.0-0(isNatKind.1(0.), n__isNatIListKind.0(activate.0(y1)))
ACTIVATE.0(n__and.0-0(n__and.0-0(x0, x1), y1)) → AND.0-0(and.0-0(activate.0(x0), x1), y1)
ACTIVATE.0(n__isNatIListKind.0(X)) → ISNATILISTKIND.0(X)

The TRS R consists of the following rules:

U12.0(tt.) → tt.
U61.0-1(tt., L) → s.1(length.1(activate.1(L)))
length.0(cons.0-1(N, L)) → U61.0-1(and.0-0(and.0-0(isNatList.1(activate.1(L)), n__isNatIListKind.1(activate.1(L))), n__and.0-0(n__isNat.0(N), n__isNatKind.0(N))), activate.1(L))
U53.0(tt.) → tt.
activate.0(n__and.1-0(X1, X2)) → and.1-0(activate.1(X1), X2)
activate.1(n__0.) → 0.
isNatIList.0(n__cons.0-1(V1, V2)) → U41.0-0-1(and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.1(activate.1(V2))), activate.0(V1), activate.1(V2))
isNatIList.1(V) → U31.0-1(isNatIListKind.1(activate.1(V)), activate.1(V))
activate.0(n__zeros.) → zeros.
and.0-1(tt., X) → activate.1(X)
isNatKind.1(n__length.1(V1)) → isNatIListKind.1(activate.1(V1))
and.0-0(X1, X2) → n__and.0-0(X1, X2)
activate.0(n__s.0(X)) → s.0(activate.0(X))
isNat.0(n__s.0(V1)) → U21.0-0(isNatKind.0(activate.0(V1)), activate.0(V1))
U51.0-0-1(tt., V1, V2) → U52.0-1(isNat.0(activate.0(V1)), activate.1(V2))
U21.0-1(tt., V1) → U22.0(isNat.1(activate.1(V1)))
isNatList.0(n__cons.1-1(V1, V2)) → U51.0-1-1(and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.1(activate.1(V2))), activate.1(V1), activate.1(V2))
length.0(X) → n__length.0(X)
isNatList.0(n__cons.0-1(V1, V2)) → U51.0-0-1(and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.1(activate.1(V2))), activate.0(V1), activate.1(V2))
U52.0-1(tt., V2) → U53.0(isNatList.1(activate.1(V2)))
cons.1-0(X1, X2) → n__cons.1-0(X1, X2)
length.1(nil.) → 0.
isNatIList.0(n__cons.1-1(V1, V2)) → U41.0-1-1(and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.1(activate.1(V2))), activate.1(V1), activate.1(V2))
and.1-1(X1, X2) → n__and.1-1(X1, X2)
activate.0(n__isNatKind.0(X)) → isNatKind.0(X)
activate.0(n__s.1(X)) → s.1(activate.1(X))
U31.0-0(tt., V) → U32.0(isNatList.0(activate.0(V)))
isNatIListKind.0(n__cons.0-0(V1, V2)) → and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.0(activate.0(V2)))
isNat.0(X) → n__isNat.0(X)
length.0(cons.0-0(N, L)) → U61.0-0(and.0-0(and.0-0(isNatList.0(activate.0(L)), n__isNatIListKind.0(activate.0(L))), n__and.0-0(n__isNat.0(N), n__isNatKind.0(N))), activate.0(L))
activate.0(n__and.0-0(X1, X2)) → and.0-0(activate.0(X1), X2)
zeros. → cons.1-0(0., n__zeros.)
activate.0(n__isNat.1(X)) → isNat.1(X)
activate.0(n__isNatIListKind.0(X)) → isNatIListKind.0(X)
cons.0-0(X1, X2) → n__cons.0-0(X1, X2)
isNatIListKind.0(n__cons.1-1(V1, V2)) → and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.1(activate.1(V2)))
s.0(X) → n__s.0(X)
cons.0-1(X1, X2) → n__cons.0-1(X1, X2)
U41.0-1-0(tt., V1, V2) → U42.0-0(isNat.1(activate.1(V1)), activate.0(V2))
isNatKind.0(X) → n__isNatKind.0(X)
activate.0(n__cons.0-1(X1, X2)) → cons.0-1(activate.0(X1), X2)
isNat.1(n__length.1(V1)) → U11.0-1(isNatIListKind.1(activate.1(V1)), activate.1(V1))
length.0(cons.1-1(N, L)) → U61.0-1(and.0-0(and.0-0(isNatList.1(activate.1(L)), n__isNatIListKind.1(activate.1(L))), n__and.0-0(n__isNat.1(N), n__isNatKind.1(N))), activate.1(L))
isNat.1(n__0.) → tt.
and.0-1(X1, X2) → n__and.0-1(X1, X2)
U21.0-0(tt., V1) → U22.0(isNat.0(activate.0(V1)))
isNatKind.0(n__length.0(V1)) → isNatIListKind.0(activate.0(V1))
zeros. → n__zeros.
U11.0-1(tt., V1) → U12.0(isNatList.1(activate.1(V1)))
U11.0-0(tt., V1) → U12.0(isNatList.0(activate.0(V1)))
U43.0(tt.) → tt.
activate.0(n__isNatKind.1(X)) → isNatKind.1(X)
isNat.1(X) → n__isNat.1(X)
activate.1(n__and.1-1(X1, X2)) → and.1-1(activate.1(X1), X2)
U31.0-1(tt., V) → U32.0(isNatList.1(activate.1(V)))
activate.0(n__cons.0-0(X1, X2)) → cons.0-0(activate.0(X1), X2)
U41.0-0-0(tt., V1, V2) → U42.0-0(isNat.0(activate.0(V1)), activate.0(V2))
U32.0(tt.) → tt.
isNat.0(n__length.0(V1)) → U11.0-0(isNatIListKind.0(activate.0(V1)), activate.0(V1))
isNatIListKind.0(X) → n__isNatIListKind.0(X)
isNat.0(n__s.1(V1)) → U21.0-1(isNatKind.1(activate.1(V1)), activate.1(V1))
s.1(X) → n__s.1(X)
U22.0(tt.) → tt.
isNatIListKind.1(X) → n__isNatIListKind.1(X)
U51.0-0-0(tt., V1, V2) → U52.0-0(isNat.0(activate.0(V1)), activate.0(V2))
isNatIListKind.0(n__zeros.) → tt.
activate.1(n__nil.) → nil.
isNatKind.0(n__s.0(V1)) → isNatKind.0(activate.0(V1))
isNatIList.0(n__cons.0-0(V1, V2)) → U41.0-0-0(and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.0(activate.0(V2))), activate.0(V1), activate.0(V2))
activate.0(n__cons.1-1(X1, X2)) → cons.1-1(activate.1(X1), X2)
isNatIList.0(n__cons.1-0(V1, V2)) → U41.0-1-0(and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.0(activate.0(V2))), activate.1(V1), activate.0(V2))
and.0-0(tt., X) → activate.0(X)
U42.0-1(tt., V2) → U43.0(isNatIList.1(activate.1(V2)))
length.1(X) → n__length.1(X)
cons.1-1(X1, X2) → n__cons.1-1(X1, X2)
and.1-0(X1, X2) → n__and.1-0(X1, X2)
activate.0(X) → X
U61.0-0(tt., L) → s.0(length.0(activate.0(L)))
activate.0(n__isNat.0(X)) → isNat.0(X)
activate.1(X) → X
isNatIListKind.1(n__nil.) → tt.
isNatKind.1(n__0.) → tt.
0. → n__0.
U41.0-1-1(tt., V1, V2) → U42.0-1(isNat.1(activate.1(V1)), activate.1(V2))
length.0(cons.1-0(N, L)) → U61.0-0(and.0-0(and.0-0(isNatList.0(activate.0(L)), n__isNatIListKind.0(activate.0(L))), n__and.0-0(n__isNat.1(N), n__isNatKind.1(N))), activate.0(L))
isNatKind.1(X) → n__isNatKind.1(X)
isNatIListKind.0(n__cons.1-0(V1, V2)) → and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.0(activate.0(V2)))
isNatIListKind.0(n__cons.0-1(V1, V2)) → and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.1(activate.1(V2)))
isNatIList.0(V) → U31.0-0(isNatIListKind.0(activate.0(V)), activate.0(V))
U42.0-0(tt., V2) → U43.0(isNatIList.0(activate.0(V2)))
isNatList.0(n__cons.1-0(V1, V2)) → U51.0-1-0(and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.0(activate.0(V2))), activate.1(V1), activate.0(V2))
activate.0(n__cons.1-0(X1, X2)) → cons.1-0(activate.1(X1), X2)
activate.1(n__length.1(X)) → length.1(activate.1(X))
isNatIList.0(n__zeros.) → tt.
activate.0(n__length.0(X)) → length.0(activate.0(X))
activate.1(n__and.0-1(X1, X2)) → and.0-1(activate.0(X1), X2)
U51.0-1-1(tt., V1, V2) → U52.0-1(isNat.1(activate.1(V1)), activate.1(V2))
U41.0-0-1(tt., V1, V2) → U42.0-1(isNat.0(activate.0(V1)), activate.1(V2))
U52.0-0(tt., V2) → U53.0(isNatList.0(activate.0(V2)))
U51.0-1-0(tt., V1, V2) → U52.0-0(isNat.1(activate.1(V1)), activate.0(V2))
isNatKind.0(n__s.1(V1)) → isNatKind.1(activate.1(V1))
isNatList.0(n__cons.0-0(V1, V2)) → U51.0-0-0(and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.0(activate.0(V2))), activate.0(V1), activate.0(V2))
nil. → n__nil.
isNatList.1(n__nil.) → tt.
activate.0(n__isNatIListKind.1(X)) → isNatIListKind.1(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.

ACTIVATE.0(n__and.0-0(x0, y1)) → AND.0-0(x0, y1)
ACTIVATE.0(n__and.0-0(X1, X2)) → ACTIVATE.0(X1)
ACTIVATE.0(n__and.0-0(n__isNatKind.1(x0), y1)) → AND.0-0(isNatKind.1(x0), y1)
ACTIVATE.0(n__s.0(X)) → ACTIVATE.0(X)
ACTIVATE.0(n__and.0-0(n__and.0-0(x0, x1), y1)) → AND.0-0(and.0-0(activate.0(x0), x1), y1)

The remaining pairs can at least be oriented weakly.

AND.0-0(tt., X) → ACTIVATE.0(X)
ISNATILISTKIND.0(n__cons.1-0(x0, y1)) → AND.0-0(isNatKind.1(x0), n__isNatIListKind.0(activate.0(y1)))
ISNATILISTKIND.0(n__cons.1-0(n__and.1-1(x0, x1), y1)) → AND.0-0(isNatKind.1(and.1-1(activate.1(x0), x1)), n__isNatIListKind.0(activate.0(y1)))
ISNATILISTKIND.0(n__cons.1-0(n__and.0-1(x0, x1), y1)) → AND.0-0(isNatKind.1(and.0-1(activate.0(x0), x1)), n__isNatIListKind.0(activate.0(y1)))
ISNATILISTKIND.0(n__cons.1-0(n__0., y1)) → AND.0-0(isNatKind.1(0.), n__isNatIListKind.0(activate.0(y1)))
ACTIVATE.0(n__isNatIListKind.0(X)) → ISNATILISTKIND.0(X)

Used ordering: Polynomial interpretation [25]:

POL(0.) = 0
POL(ACTIVATE.0(x₁)) = x₁
POL(AND.0-0(x₁, x₂)) = x₂
POL(ISNATILISTKIND.0(x₁)) = 0
POL(U11.0-0(x₁, x₂)) = 0
POL(U11.0-1(x₁, x₂)) = 0
POL(U12.0(x₁)) = 0
POL(U21.0-0(x₁, x₂)) = 0
POL(U21.0-1(x₁, x₂)) = 0
POL(U22.0(x₁)) = 0
POL(U51.0-0-0(x₁, x₂, x₃)) = 0
POL(U51.0-0-1(x₁, x₂, x₃)) = 0
POL(U51.0-1-0(x₁, x₂, x₃)) = 0
POL(U51.0-1-1(x₁, x₂, x₃)) = 0
POL(U52.0-0(x₁, x₂)) = 0
POL(U52.0-1(x₁, x₂)) = 0
POL(U53.0(x₁)) = 0
POL(U61.0-0(x₁, x₂)) = 0
POL(U61.0-1(x₁, x₂)) = 0
POL(activate.0(x₁)) = 0
POL(activate.1(x₁)) = 0
POL(and.0-0(x₁, x₂)) = 0
POL(and.0-1(x₁, x₂)) = 0
POL(and.1-0(x₁, x₂)) = 0
POL(and.1-1(x₁, x₂)) = 0
POL(cons.0-0(x₁, x₂)) = 0
POL(cons.0-1(x₁, x₂)) = 0
POL(cons.1-0(x₁, x₂)) = 0
POL(cons.1-1(x₁, x₂)) = 0
POL(isNat.0(x₁)) = 0
POL(isNat.1(x₁)) = 0
POL(isNatIListKind.0(x₁)) = 0
POL(isNatIListKind.1(x₁)) = 0
POL(isNatKind.0(x₁)) = 0
POL(isNatKind.1(x₁)) = 0
POL(isNatList.0(x₁)) = 0
POL(isNatList.1(x₁)) = 1
POL(length.0(x₁)) = 0
POL(length.1(x₁)) = 0
POL(n__0.) = 0
POL(n__and.0-0(x₁, x₂)) = 1 + x₁ + x₂
POL(n__and.0-1(x₁, x₂)) = 0
POL(n__and.1-0(x₁, x₂)) = 0
POL(n__and.1-1(x₁, x₂)) = 0
POL(n__cons.0-0(x₁, x₂)) = 0
POL(n__cons.0-1(x₁, x₂)) = 0
POL(n__cons.1-0(x₁, x₂)) = 0
POL(n__cons.1-1(x₁, x₂)) = 0
POL(n__isNat.0(x₁)) = 0
POL(n__isNat.1(x₁)) = 0
POL(n__isNatIListKind.0(x₁)) = 0
POL(n__isNatIListKind.1(x₁)) = 0
POL(n__isNatKind.0(x₁)) = 0
POL(n__isNatKind.1(x₁)) = 1
POL(n__length.0(x₁)) = 0
POL(n__length.1(x₁)) = 0
POL(n__nil.) = 0
POL(n__s.0(x₁)) = 1 + x₁
POL(n__s.1(x₁)) = 0
POL(n__zeros.) = 0
POL(nil.) = 0
POL(s.0(x₁)) = 0
POL(s.1(x₁)) = 0
POL(tt.) = 1
POL(zeros.) = 0

The following usable rules [17] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ 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

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ SemLabProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                              ↳ AND

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ QDPOrderProof

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                            ↳ SemLabProof2

                              ↳ QDP

                    ↳ QDP

          ↳ QDP

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

AND.0-0(tt., X) → ACTIVATE.0(X)
ISNATILISTKIND.0(n__cons.1-0(x0, y1)) → AND.0-0(isNatKind.1(x0), n__isNatIListKind.0(activate.0(y1)))
ISNATILISTKIND.0(n__cons.1-0(n__and.1-1(x0, x1), y1)) → AND.0-0(isNatKind.1(and.1-1(activate.1(x0), x1)), n__isNatIListKind.0(activate.0(y1)))
ISNATILISTKIND.0(n__cons.1-0(n__and.0-1(x0, x1), y1)) → AND.0-0(isNatKind.1(and.0-1(activate.0(x0), x1)), n__isNatIListKind.0(activate.0(y1)))
ISNATILISTKIND.0(n__cons.1-0(n__0., y1)) → AND.0-0(isNatKind.1(0.), n__isNatIListKind.0(activate.0(y1)))
ACTIVATE.0(n__isNatIListKind.0(X)) → ISNATILISTKIND.0(X)

The TRS R consists of the following rules:

U12.0(tt.) → tt.
U61.0-1(tt., L) → s.1(length.1(activate.1(L)))
length.0(cons.0-1(N, L)) → U61.0-1(and.0-0(and.0-0(isNatList.1(activate.1(L)), n__isNatIListKind.1(activate.1(L))), n__and.0-0(n__isNat.0(N), n__isNatKind.0(N))), activate.1(L))
U53.0(tt.) → tt.
activate.0(n__and.1-0(X1, X2)) → and.1-0(activate.1(X1), X2)
activate.1(n__0.) → 0.
isNatIList.0(n__cons.0-1(V1, V2)) → U41.0-0-1(and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.1(activate.1(V2))), activate.0(V1), activate.1(V2))
isNatIList.1(V) → U31.0-1(isNatIListKind.1(activate.1(V)), activate.1(V))
activate.0(n__zeros.) → zeros.
and.0-1(tt., X) → activate.1(X)
isNatKind.1(n__length.1(V1)) → isNatIListKind.1(activate.1(V1))
and.0-0(X1, X2) → n__and.0-0(X1, X2)
activate.0(n__s.0(X)) → s.0(activate.0(X))
isNat.0(n__s.0(V1)) → U21.0-0(isNatKind.0(activate.0(V1)), activate.0(V1))
U51.0-0-1(tt., V1, V2) → U52.0-1(isNat.0(activate.0(V1)), activate.1(V2))
U21.0-1(tt., V1) → U22.0(isNat.1(activate.1(V1)))
isNatList.0(n__cons.1-1(V1, V2)) → U51.0-1-1(and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.1(activate.1(V2))), activate.1(V1), activate.1(V2))
length.0(X) → n__length.0(X)
isNatList.0(n__cons.0-1(V1, V2)) → U51.0-0-1(and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.1(activate.1(V2))), activate.0(V1), activate.1(V2))
U52.0-1(tt., V2) → U53.0(isNatList.1(activate.1(V2)))
cons.1-0(X1, X2) → n__cons.1-0(X1, X2)
length.1(nil.) → 0.
isNatIList.0(n__cons.1-1(V1, V2)) → U41.0-1-1(and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.1(activate.1(V2))), activate.1(V1), activate.1(V2))
and.1-1(X1, X2) → n__and.1-1(X1, X2)
activate.0(n__isNatKind.0(X)) → isNatKind.0(X)
activate.0(n__s.1(X)) → s.1(activate.1(X))
U31.0-0(tt., V) → U32.0(isNatList.0(activate.0(V)))
isNatIListKind.0(n__cons.0-0(V1, V2)) → and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.0(activate.0(V2)))
isNat.0(X) → n__isNat.0(X)
length.0(cons.0-0(N, L)) → U61.0-0(and.0-0(and.0-0(isNatList.0(activate.0(L)), n__isNatIListKind.0(activate.0(L))), n__and.0-0(n__isNat.0(N), n__isNatKind.0(N))), activate.0(L))
activate.0(n__and.0-0(X1, X2)) → and.0-0(activate.0(X1), X2)
zeros. → cons.1-0(0., n__zeros.)
activate.0(n__isNat.1(X)) → isNat.1(X)
activate.0(n__isNatIListKind.0(X)) → isNatIListKind.0(X)
cons.0-0(X1, X2) → n__cons.0-0(X1, X2)
isNatIListKind.0(n__cons.1-1(V1, V2)) → and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.1(activate.1(V2)))
s.0(X) → n__s.0(X)
cons.0-1(X1, X2) → n__cons.0-1(X1, X2)
U41.0-1-0(tt., V1, V2) → U42.0-0(isNat.1(activate.1(V1)), activate.0(V2))
isNatKind.0(X) → n__isNatKind.0(X)
activate.0(n__cons.0-1(X1, X2)) → cons.0-1(activate.0(X1), X2)
isNat.1(n__length.1(V1)) → U11.0-1(isNatIListKind.1(activate.1(V1)), activate.1(V1))
length.0(cons.1-1(N, L)) → U61.0-1(and.0-0(and.0-0(isNatList.1(activate.1(L)), n__isNatIListKind.1(activate.1(L))), n__and.0-0(n__isNat.1(N), n__isNatKind.1(N))), activate.1(L))
isNat.1(n__0.) → tt.
and.0-1(X1, X2) → n__and.0-1(X1, X2)
U21.0-0(tt., V1) → U22.0(isNat.0(activate.0(V1)))
isNatKind.0(n__length.0(V1)) → isNatIListKind.0(activate.0(V1))
zeros. → n__zeros.
U11.0-1(tt., V1) → U12.0(isNatList.1(activate.1(V1)))
U11.0-0(tt., V1) → U12.0(isNatList.0(activate.0(V1)))
U43.0(tt.) → tt.
activate.0(n__isNatKind.1(X)) → isNatKind.1(X)
isNat.1(X) → n__isNat.1(X)
activate.1(n__and.1-1(X1, X2)) → and.1-1(activate.1(X1), X2)
U31.0-1(tt., V) → U32.0(isNatList.1(activate.1(V)))
activate.0(n__cons.0-0(X1, X2)) → cons.0-0(activate.0(X1), X2)
U41.0-0-0(tt., V1, V2) → U42.0-0(isNat.0(activate.0(V1)), activate.0(V2))
U32.0(tt.) → tt.
isNat.0(n__length.0(V1)) → U11.0-0(isNatIListKind.0(activate.0(V1)), activate.0(V1))
isNatIListKind.0(X) → n__isNatIListKind.0(X)
isNat.0(n__s.1(V1)) → U21.0-1(isNatKind.1(activate.1(V1)), activate.1(V1))
s.1(X) → n__s.1(X)
U22.0(tt.) → tt.
isNatIListKind.1(X) → n__isNatIListKind.1(X)
U51.0-0-0(tt., V1, V2) → U52.0-0(isNat.0(activate.0(V1)), activate.0(V2))
isNatIListKind.0(n__zeros.) → tt.
activate.1(n__nil.) → nil.
isNatKind.0(n__s.0(V1)) → isNatKind.0(activate.0(V1))
isNatIList.0(n__cons.0-0(V1, V2)) → U41.0-0-0(and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.0(activate.0(V2))), activate.0(V1), activate.0(V2))
activate.0(n__cons.1-1(X1, X2)) → cons.1-1(activate.1(X1), X2)
isNatIList.0(n__cons.1-0(V1, V2)) → U41.0-1-0(and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.0(activate.0(V2))), activate.1(V1), activate.0(V2))
and.0-0(tt., X) → activate.0(X)
U42.0-1(tt., V2) → U43.0(isNatIList.1(activate.1(V2)))
length.1(X) → n__length.1(X)
cons.1-1(X1, X2) → n__cons.1-1(X1, X2)
and.1-0(X1, X2) → n__and.1-0(X1, X2)
activate.0(X) → X
U61.0-0(tt., L) → s.0(length.0(activate.0(L)))
activate.0(n__isNat.0(X)) → isNat.0(X)
activate.1(X) → X
isNatIListKind.1(n__nil.) → tt.
isNatKind.1(n__0.) → tt.
0. → n__0.
U41.0-1-1(tt., V1, V2) → U42.0-1(isNat.1(activate.1(V1)), activate.1(V2))
length.0(cons.1-0(N, L)) → U61.0-0(and.0-0(and.0-0(isNatList.0(activate.0(L)), n__isNatIListKind.0(activate.0(L))), n__and.0-0(n__isNat.1(N), n__isNatKind.1(N))), activate.0(L))
isNatKind.1(X) → n__isNatKind.1(X)
isNatIListKind.0(n__cons.1-0(V1, V2)) → and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.0(activate.0(V2)))
isNatIListKind.0(n__cons.0-1(V1, V2)) → and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.1(activate.1(V2)))
isNatIList.0(V) → U31.0-0(isNatIListKind.0(activate.0(V)), activate.0(V))
U42.0-0(tt., V2) → U43.0(isNatIList.0(activate.0(V2)))
isNatList.0(n__cons.1-0(V1, V2)) → U51.0-1-0(and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.0(activate.0(V2))), activate.1(V1), activate.0(V2))
activate.0(n__cons.1-0(X1, X2)) → cons.1-0(activate.1(X1), X2)
activate.1(n__length.1(X)) → length.1(activate.1(X))
isNatIList.0(n__zeros.) → tt.
activate.0(n__length.0(X)) → length.0(activate.0(X))
activate.1(n__and.0-1(X1, X2)) → and.0-1(activate.0(X1), X2)
U51.0-1-1(tt., V1, V2) → U52.0-1(isNat.1(activate.1(V1)), activate.1(V2))
U41.0-0-1(tt., V1, V2) → U42.0-1(isNat.0(activate.0(V1)), activate.1(V2))
U52.0-0(tt., V2) → U53.0(isNatList.0(activate.0(V2)))
U51.0-1-0(tt., V1, V2) → U52.0-0(isNat.1(activate.1(V1)), activate.0(V2))
isNatKind.0(n__s.1(V1)) → isNatKind.1(activate.1(V1))
isNatList.0(n__cons.0-0(V1, V2)) → U51.0-0-0(and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.0(activate.0(V2))), activate.0(V1), activate.0(V2))
nil. → n__nil.
isNatList.1(n__nil.) → tt.
activate.0(n__isNatIListKind.1(X)) → isNatIListKind.1(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.0(n__cons.1-0(n__and.0-1(x0, x1), y1)) → AND.0-0(isNatKind.1(and.0-1(activate.0(x0), x1)), n__isNatIListKind.0(activate.0(y1)))

The remaining pairs can at least be oriented weakly.

AND.0-0(tt., X) → ACTIVATE.0(X)
ISNATILISTKIND.0(n__cons.1-0(x0, y1)) → AND.0-0(isNatKind.1(x0), n__isNatIListKind.0(activate.0(y1)))
ISNATILISTKIND.0(n__cons.1-0(n__and.1-1(x0, x1), y1)) → AND.0-0(isNatKind.1(and.1-1(activate.1(x0), x1)), n__isNatIListKind.0(activate.0(y1)))
ISNATILISTKIND.0(n__cons.1-0(n__0., y1)) → AND.0-0(isNatKind.1(0.), n__isNatIListKind.0(activate.0(y1)))
ACTIVATE.0(n__isNatIListKind.0(X)) → ISNATILISTKIND.0(X)

Used ordering: Polynomial interpretation [25]:

POL(0.) = 0
POL(ACTIVATE.0(x₁)) = x₁
POL(AND.0-0(x₁, x₂)) = x₂
POL(ISNATILISTKIND.0(x₁)) = x₁
POL(U11.0-0(x₁, x₂)) = 0
POL(U11.0-1(x₁, x₂)) = 1 + x₂
POL(U12.0(x₁)) = 0
POL(U21.0-0(x₁, x₂)) = 0
POL(U21.0-1(x₁, x₂)) = 0
POL(U22.0(x₁)) = 0
POL(U51.0-0-0(x₁, x₂, x₃)) = 1
POL(U51.0-0-1(x₁, x₂, x₃)) = 0
POL(U51.0-1-0(x₁, x₂, x₃)) = 1
POL(U51.0-1-1(x₁, x₂, x₃)) = 1
POL(U52.0-0(x₁, x₂)) = 1
POL(U52.0-1(x₁, x₂)) = 0
POL(U53.0(x₁)) = 0
POL(U61.0-0(x₁, x₂)) = x₂
POL(U61.0-1(x₁, x₂)) = 0
POL(activate.0(x₁)) = x₁
POL(activate.1(x₁)) = x₁
POL(and.0-0(x₁, x₂)) = x₂
POL(and.0-1(x₁, x₂)) = 1 + x₂
POL(and.1-0(x₁, x₂)) = 0
POL(and.1-1(x₁, x₂)) = 0
POL(cons.0-0(x₁, x₂)) = x₁ + x₂
POL(cons.0-1(x₁, x₂)) = 1 + x₁
POL(cons.1-0(x₁, x₂)) = x₁ + x₂
POL(cons.1-1(x₁, x₂)) = x₁
POL(isNat.0(x₁)) = 0
POL(isNat.1(x₁)) = 1 + x₁
POL(isNatIListKind.0(x₁)) = x₁
POL(isNatIListKind.1(x₁)) = 0
POL(isNatKind.0(x₁)) = x₁
POL(isNatKind.1(x₁)) = 0
POL(isNatList.0(x₁)) = 1
POL(isNatList.1(x₁)) = 0
POL(length.0(x₁)) = x₁
POL(length.1(x₁)) = 1 + x₁
POL(n__0.) = 0
POL(n__and.0-0(x₁, x₂)) = x₂
POL(n__and.0-1(x₁, x₂)) = 1 + x₂
POL(n__and.1-0(x₁, x₂)) = 0
POL(n__and.1-1(x₁, x₂)) = 0
POL(n__cons.0-0(x₁, x₂)) = x₁ + x₂
POL(n__cons.0-1(x₁, x₂)) = 1 + x₁
POL(n__cons.1-0(x₁, x₂)) = x₁ + x₂
POL(n__cons.1-1(x₁, x₂)) = x₁
POL(n__isNat.0(x₁)) = 0
POL(n__isNat.1(x₁)) = 1 + x₁
POL(n__isNatIListKind.0(x₁)) = x₁
POL(n__isNatIListKind.1(x₁)) = 0
POL(n__isNatKind.0(x₁)) = x₁
POL(n__isNatKind.1(x₁)) = 0
POL(n__length.0(x₁)) = x₁
POL(n__length.1(x₁)) = 1 + x₁
POL(n__nil.) = 0
POL(n__s.0(x₁)) = x₁
POL(n__s.1(x₁)) = 0
POL(n__zeros.) = 0
POL(nil.) = 0
POL(s.0(x₁)) = x₁
POL(s.1(x₁)) = 0
POL(tt.) = 0
POL(zeros.) = 0

The following usable rules [17] were oriented:

length.1(X) → n__length.1(X)
cons.1-1(X1, X2) → n__cons.1-1(X1, X2)
and.1-0(X1, X2) → n__and.1-0(X1, X2)
activate.0(X) → X
U61.0-0(tt., L) → s.0(length.0(activate.0(L)))
U22.0(tt.) → tt.
isNatIListKind.1(X) → n__isNatIListKind.1(X)
U51.0-0-0(tt., V1, V2) → U52.0-0(isNat.0(activate.0(V1)), activate.0(V2))
isNatIListKind.0(n__zeros.) → tt.
activate.1(n__nil.) → nil.
activate.0(n__cons.1-1(X1, X2)) → cons.1-1(activate.1(X1), X2)
activate.0(n__cons.0-0(X1, X2)) → cons.0-0(activate.0(X1), X2)
isNatIListKind.0(X) → n__isNatIListKind.0(X)
isNat.0(n__length.0(V1)) → U11.0-0(isNatIListKind.0(activate.0(V1)), activate.0(V1))
s.1(X) → n__s.1(X)
isNat.0(n__s.1(V1)) → U21.0-1(isNatKind.1(activate.1(V1)), activate.1(V1))
zeros. → n__zeros.
U11.0-0(tt., V1) → U12.0(isNatList.0(activate.0(V1)))
U11.0-1(tt., V1) → U12.0(isNatList.1(activate.1(V1)))
activate.0(n__isNatKind.1(X)) → isNatKind.1(X)
activate.1(n__and.1-1(X1, X2)) → and.1-1(activate.1(X1), X2)
isNat.1(X) → n__isNat.1(X)
activate.0(n__isNatIListKind.1(X)) → isNatIListKind.1(X)
nil. → n__nil.
U51.0-1-1(tt., V1, V2) → U52.0-1(isNat.1(activate.1(V1)), activate.1(V2))
activate.0(n__length.0(X)) → length.0(activate.0(X))
and.0-1(tt., X) → activate.1(X)
activate.1(n__and.0-1(X1, X2)) → and.0-1(activate.0(X1), X2)
isNatKind.0(n__s.1(V1)) → isNatKind.1(activate.1(V1))
isNatList.0(n__cons.0-0(V1, V2)) → U51.0-0-0(and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.0(activate.0(V2))), activate.0(V1), activate.0(V2))
U52.0-0(tt., V2) → U53.0(isNatList.0(activate.0(V2)))
U51.0-1-0(tt., V1, V2) → U52.0-0(isNat.1(activate.1(V1)), activate.0(V2))
activate.1(n__length.1(X)) → length.1(activate.1(X))
activate.0(n__cons.1-0(X1, X2)) → cons.1-0(activate.1(X1), X2)
isNatList.0(n__cons.1-0(V1, V2)) → U51.0-1-0(and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.0(activate.0(V2))), activate.1(V1), activate.0(V2))
isNatKind.1(n__0.) → tt.
isNatIListKind.1(n__nil.) → tt.
activate.1(X) → X
activate.0(n__isNat.0(X)) → isNat.0(X)
isNatKind.1(X) → n__isNatKind.1(X)
length.0(cons.1-0(N, L)) → U61.0-0(and.0-0(and.0-0(isNatList.0(activate.0(L)), n__isNatIListKind.0(activate.0(L))), n__and.0-0(n__isNat.1(N), n__isNatKind.1(N))), activate.0(L))
0. → n__0.
isNatKind.1(n__length.1(V1)) → isNatIListKind.1(activate.1(V1))
and.0-0(X1, X2) → n__and.0-0(X1, X2)
activate.0(n__s.0(X)) → s.0(activate.0(X))
isNat.0(n__s.0(V1)) → U21.0-0(isNatKind.0(activate.0(V1)), activate.0(V1))
activate.0(n__zeros.) → zeros.
length.0(cons.0-1(N, L)) → U61.0-1(and.0-0(and.0-0(isNatList.1(activate.1(L)), n__isNatIListKind.1(activate.1(L))), n__and.0-0(n__isNat.0(N), n__isNatKind.0(N))), activate.1(L))
U53.0(tt.) → tt.
activate.0(n__and.1-0(X1, X2)) → and.1-0(activate.1(X1), X2)
activate.1(n__0.) → 0.
U12.0(tt.) → tt.
U61.0-1(tt., L) → s.1(length.1(activate.1(L)))
and.0-1(X1, X2) → n__and.0-1(X1, X2)
U21.0-0(tt., V1) → U22.0(isNat.0(activate.0(V1)))
length.0(cons.1-1(N, L)) → U61.0-1(and.0-0(and.0-0(isNatList.1(activate.1(L)), n__isNatIListKind.1(activate.1(L))), n__and.0-0(n__isNat.1(N), n__isNatKind.1(N))), activate.1(L))
isNat.1(n__0.) → tt.
activate.0(n__cons.0-1(X1, X2)) → cons.0-1(activate.0(X1), X2)
isNat.1(n__length.1(V1)) → U11.0-1(isNatIListKind.1(activate.1(V1)), activate.1(V1))
isNatKind.0(X) → n__isNatKind.0(X)
s.0(X) → n__s.0(X)
cons.0-1(X1, X2) → n__cons.0-1(X1, X2)
cons.0-0(X1, X2) → n__cons.0-0(X1, X2)
activate.0(n__isNat.1(X)) → isNat.1(X)
zeros. → cons.1-0(0., n__zeros.)
length.0(cons.0-0(N, L)) → U61.0-0(and.0-0(and.0-0(isNatList.0(activate.0(L)), n__isNatIListKind.0(activate.0(L))), n__and.0-0(n__isNat.0(N), n__isNatKind.0(N))), activate.0(L))
isNat.0(X) → n__isNat.0(X)
activate.0(n__s.1(X)) → s.1(activate.1(X))
isNatIListKind.0(n__cons.1-0(V1, V2)) → and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.0(activate.0(V2)))
isNatIListKind.0(n__cons.1-1(V1, V2)) → and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.1(activate.1(V2)))
isNatIListKind.0(n__cons.0-0(V1, V2)) → and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.0(activate.0(V2)))
isNatIListKind.0(n__cons.0-1(V1, V2)) → and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.1(activate.1(V2)))
and.0-0(tt., X) → activate.0(X)
activate.0(n__and.0-0(X1, X2)) → and.0-0(activate.0(X1), X2)
isNatKind.0(n__s.0(V1)) → isNatKind.0(activate.0(V1))
isNatKind.0(n__length.0(V1)) → isNatIListKind.0(activate.0(V1))
activate.0(n__isNatIListKind.0(X)) → isNatIListKind.0(X)
activate.0(n__isNatKind.0(X)) → isNatKind.0(X)
and.1-1(X1, X2) → n__and.1-1(X1, X2)
length.1(nil.) → 0.
cons.1-0(X1, X2) → n__cons.1-0(X1, X2)
U52.0-1(tt., V2) → U53.0(isNatList.1(activate.1(V2)))
isNatList.0(n__cons.0-1(V1, V2)) → U51.0-0-1(and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.1(activate.1(V2))), activate.0(V1), activate.1(V2))
length.0(X) → n__length.0(X)
isNatList.0(n__cons.1-1(V1, V2)) → U51.0-1-1(and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.1(activate.1(V2))), activate.1(V1), activate.1(V2))
U21.0-1(tt., V1) → U22.0(isNat.1(activate.1(V1)))
U51.0-0-1(tt., V1, V2) → U52.0-1(isNat.0(activate.0(V1)), activate.1(V2))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ 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

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ SemLabProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                              ↳ AND

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ QDPOrderProof

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ QDPOrderProof

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                            ↳ SemLabProof2

                              ↳ QDP

                    ↳ QDP

          ↳ QDP

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

AND.0-0(tt., X) → ACTIVATE.0(X)
ISNATILISTKIND.0(n__cons.1-0(x0, y1)) → AND.0-0(isNatKind.1(x0), n__isNatIListKind.0(activate.0(y1)))
ISNATILISTKIND.0(n__cons.1-0(n__and.1-1(x0, x1), y1)) → AND.0-0(isNatKind.1(and.1-1(activate.1(x0), x1)), n__isNatIListKind.0(activate.0(y1)))
ISNATILISTKIND.0(n__cons.1-0(n__0., y1)) → AND.0-0(isNatKind.1(0.), n__isNatIListKind.0(activate.0(y1)))
ACTIVATE.0(n__isNatIListKind.0(X)) → ISNATILISTKIND.0(X)

The TRS R consists of the following rules:

U12.0(tt.) → tt.
U61.0-1(tt., L) → s.1(length.1(activate.1(L)))
length.0(cons.0-1(N, L)) → U61.0-1(and.0-0(and.0-0(isNatList.1(activate.1(L)), n__isNatIListKind.1(activate.1(L))), n__and.0-0(n__isNat.0(N), n__isNatKind.0(N))), activate.1(L))
U53.0(tt.) → tt.
activate.0(n__and.1-0(X1, X2)) → and.1-0(activate.1(X1), X2)
activate.1(n__0.) → 0.
isNatIList.0(n__cons.0-1(V1, V2)) → U41.0-0-1(and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.1(activate.1(V2))), activate.0(V1), activate.1(V2))
isNatIList.1(V) → U31.0-1(isNatIListKind.1(activate.1(V)), activate.1(V))
activate.0(n__zeros.) → zeros.
and.0-1(tt., X) → activate.1(X)
isNatKind.1(n__length.1(V1)) → isNatIListKind.1(activate.1(V1))
and.0-0(X1, X2) → n__and.0-0(X1, X2)
activate.0(n__s.0(X)) → s.0(activate.0(X))
isNat.0(n__s.0(V1)) → U21.0-0(isNatKind.0(activate.0(V1)), activate.0(V1))
U51.0-0-1(tt., V1, V2) → U52.0-1(isNat.0(activate.0(V1)), activate.1(V2))
U21.0-1(tt., V1) → U22.0(isNat.1(activate.1(V1)))
isNatList.0(n__cons.1-1(V1, V2)) → U51.0-1-1(and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.1(activate.1(V2))), activate.1(V1), activate.1(V2))
length.0(X) → n__length.0(X)
isNatList.0(n__cons.0-1(V1, V2)) → U51.0-0-1(and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.1(activate.1(V2))), activate.0(V1), activate.1(V2))
U52.0-1(tt., V2) → U53.0(isNatList.1(activate.1(V2)))
cons.1-0(X1, X2) → n__cons.1-0(X1, X2)
length.1(nil.) → 0.
isNatIList.0(n__cons.1-1(V1, V2)) → U41.0-1-1(and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.1(activate.1(V2))), activate.1(V1), activate.1(V2))
and.1-1(X1, X2) → n__and.1-1(X1, X2)
activate.0(n__isNatKind.0(X)) → isNatKind.0(X)
activate.0(n__s.1(X)) → s.1(activate.1(X))
U31.0-0(tt., V) → U32.0(isNatList.0(activate.0(V)))
isNatIListKind.0(n__cons.0-0(V1, V2)) → and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.0(activate.0(V2)))
isNat.0(X) → n__isNat.0(X)
length.0(cons.0-0(N, L)) → U61.0-0(and.0-0(and.0-0(isNatList.0(activate.0(L)), n__isNatIListKind.0(activate.0(L))), n__and.0-0(n__isNat.0(N), n__isNatKind.0(N))), activate.0(L))
activate.0(n__and.0-0(X1, X2)) → and.0-0(activate.0(X1), X2)
zeros. → cons.1-0(0., n__zeros.)
activate.0(n__isNat.1(X)) → isNat.1(X)
activate.0(n__isNatIListKind.0(X)) → isNatIListKind.0(X)
cons.0-0(X1, X2) → n__cons.0-0(X1, X2)
isNatIListKind.0(n__cons.1-1(V1, V2)) → and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.1(activate.1(V2)))
s.0(X) → n__s.0(X)
cons.0-1(X1, X2) → n__cons.0-1(X1, X2)
U41.0-1-0(tt., V1, V2) → U42.0-0(isNat.1(activate.1(V1)), activate.0(V2))
isNatKind.0(X) → n__isNatKind.0(X)
activate.0(n__cons.0-1(X1, X2)) → cons.0-1(activate.0(X1), X2)
isNat.1(n__length.1(V1)) → U11.0-1(isNatIListKind.1(activate.1(V1)), activate.1(V1))
length.0(cons.1-1(N, L)) → U61.0-1(and.0-0(and.0-0(isNatList.1(activate.1(L)), n__isNatIListKind.1(activate.1(L))), n__and.0-0(n__isNat.1(N), n__isNatKind.1(N))), activate.1(L))
isNat.1(n__0.) → tt.
and.0-1(X1, X2) → n__and.0-1(X1, X2)
U21.0-0(tt., V1) → U22.0(isNat.0(activate.0(V1)))
isNatKind.0(n__length.0(V1)) → isNatIListKind.0(activate.0(V1))
zeros. → n__zeros.
U11.0-1(tt., V1) → U12.0(isNatList.1(activate.1(V1)))
U11.0-0(tt., V1) → U12.0(isNatList.0(activate.0(V1)))
U43.0(tt.) → tt.
activate.0(n__isNatKind.1(X)) → isNatKind.1(X)
isNat.1(X) → n__isNat.1(X)
activate.1(n__and.1-1(X1, X2)) → and.1-1(activate.1(X1), X2)
U31.0-1(tt., V) → U32.0(isNatList.1(activate.1(V)))
activate.0(n__cons.0-0(X1, X2)) → cons.0-0(activate.0(X1), X2)
U41.0-0-0(tt., V1, V2) → U42.0-0(isNat.0(activate.0(V1)), activate.0(V2))
U32.0(tt.) → tt.
isNat.0(n__length.0(V1)) → U11.0-0(isNatIListKind.0(activate.0(V1)), activate.0(V1))
isNatIListKind.0(X) → n__isNatIListKind.0(X)
isNat.0(n__s.1(V1)) → U21.0-1(isNatKind.1(activate.1(V1)), activate.1(V1))
s.1(X) → n__s.1(X)
U22.0(tt.) → tt.
isNatIListKind.1(X) → n__isNatIListKind.1(X)
U51.0-0-0(tt., V1, V2) → U52.0-0(isNat.0(activate.0(V1)), activate.0(V2))
isNatIListKind.0(n__zeros.) → tt.
activate.1(n__nil.) → nil.
isNatKind.0(n__s.0(V1)) → isNatKind.0(activate.0(V1))
isNatIList.0(n__cons.0-0(V1, V2)) → U41.0-0-0(and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.0(activate.0(V2))), activate.0(V1), activate.0(V2))
activate.0(n__cons.1-1(X1, X2)) → cons.1-1(activate.1(X1), X2)
isNatIList.0(n__cons.1-0(V1, V2)) → U41.0-1-0(and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.0(activate.0(V2))), activate.1(V1), activate.0(V2))
and.0-0(tt., X) → activate.0(X)
U42.0-1(tt., V2) → U43.0(isNatIList.1(activate.1(V2)))
length.1(X) → n__length.1(X)
cons.1-1(X1, X2) → n__cons.1-1(X1, X2)
and.1-0(X1, X2) → n__and.1-0(X1, X2)
activate.0(X) → X
U61.0-0(tt., L) → s.0(length.0(activate.0(L)))
activate.0(n__isNat.0(X)) → isNat.0(X)
activate.1(X) → X
isNatIListKind.1(n__nil.) → tt.
isNatKind.1(n__0.) → tt.
0. → n__0.
U41.0-1-1(tt., V1, V2) → U42.0-1(isNat.1(activate.1(V1)), activate.1(V2))
length.0(cons.1-0(N, L)) → U61.0-0(and.0-0(and.0-0(isNatList.0(activate.0(L)), n__isNatIListKind.0(activate.0(L))), n__and.0-0(n__isNat.1(N), n__isNatKind.1(N))), activate.0(L))
isNatKind.1(X) → n__isNatKind.1(X)
isNatIListKind.0(n__cons.1-0(V1, V2)) → and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.0(activate.0(V2)))
isNatIListKind.0(n__cons.0-1(V1, V2)) → and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.1(activate.1(V2)))
isNatIList.0(V) → U31.0-0(isNatIListKind.0(activate.0(V)), activate.0(V))
U42.0-0(tt., V2) → U43.0(isNatIList.0(activate.0(V2)))
isNatList.0(n__cons.1-0(V1, V2)) → U51.0-1-0(and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.0(activate.0(V2))), activate.1(V1), activate.0(V2))
activate.0(n__cons.1-0(X1, X2)) → cons.1-0(activate.1(X1), X2)
activate.1(n__length.1(X)) → length.1(activate.1(X))
isNatIList.0(n__zeros.) → tt.
activate.0(n__length.0(X)) → length.0(activate.0(X))
activate.1(n__and.0-1(X1, X2)) → and.0-1(activate.0(X1), X2)
U51.0-1-1(tt., V1, V2) → U52.0-1(isNat.1(activate.1(V1)), activate.1(V2))
U41.0-0-1(tt., V1, V2) → U42.0-1(isNat.0(activate.0(V1)), activate.1(V2))
U52.0-0(tt., V2) → U53.0(isNatList.0(activate.0(V2)))
U51.0-1-0(tt., V1, V2) → U52.0-0(isNat.1(activate.1(V1)), activate.0(V2))
isNatKind.0(n__s.1(V1)) → isNatKind.1(activate.1(V1))
isNatList.0(n__cons.0-0(V1, V2)) → U51.0-0-0(and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.0(activate.0(V2))), activate.0(V1), activate.0(V2))
nil. → n__nil.
isNatList.1(n__nil.) → tt.
activate.0(n__isNatIListKind.1(X)) → isNatIListKind.1(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.0(n__cons.1-0(n__and.1-1(x0, x1), y1)) → AND.0-0(isNatKind.1(and.1-1(activate.1(x0), x1)), n__isNatIListKind.0(activate.0(y1)))

The remaining pairs can at least be oriented weakly.

AND.0-0(tt., X) → ACTIVATE.0(X)
ISNATILISTKIND.0(n__cons.1-0(x0, y1)) → AND.0-0(isNatKind.1(x0), n__isNatIListKind.0(activate.0(y1)))
ISNATILISTKIND.0(n__cons.1-0(n__0., y1)) → AND.0-0(isNatKind.1(0.), n__isNatIListKind.0(activate.0(y1)))
ACTIVATE.0(n__isNatIListKind.0(X)) → ISNATILISTKIND.0(X)

Used ordering: Polynomial interpretation [25]:

POL(0.) = 0
POL(ACTIVATE.0(x₁)) = x₁
POL(AND.0-0(x₁, x₂)) = x₂
POL(ISNATILISTKIND.0(x₁)) = x₁
POL(U11.0-0(x₁, x₂)) = x₂
POL(U11.0-1(x₁, x₂)) = 0
POL(U12.0(x₁)) = 0
POL(U21.0-0(x₁, x₂)) = 0
POL(U21.0-1(x₁, x₂)) = 0
POL(U22.0(x₁)) = 0
POL(U51.0-0-0(x₁, x₂, x₃)) = x₃
POL(U51.0-0-1(x₁, x₂, x₃)) = 0
POL(U51.0-1-0(x₁, x₂, x₃)) = x₂
POL(U51.0-1-1(x₁, x₂, x₃)) = 0
POL(U52.0-0(x₁, x₂)) = 0
POL(U52.0-1(x₁, x₂)) = 0
POL(U53.0(x₁)) = 0
POL(U61.0-0(x₁, x₂)) = x₁ + x₂
POL(U61.0-1(x₁, x₂)) = 0
POL(activate.0(x₁)) = x₁
POL(activate.1(x₁)) = x₁
POL(and.0-0(x₁, x₂)) = x₂
POL(and.0-1(x₁, x₂)) = x₁ + x₂
POL(and.1-0(x₁, x₂)) = 0
POL(and.1-1(x₁, x₂)) = 1
POL(cons.0-0(x₁, x₂)) = x₁ + x₂
POL(cons.0-1(x₁, x₂)) = 0
POL(cons.1-0(x₁, x₂)) = x₁ + x₂
POL(cons.1-1(x₁, x₂)) = 0
POL(isNat.0(x₁)) = x₁
POL(isNat.1(x₁)) = 0
POL(isNatIListKind.0(x₁)) = x₁
POL(isNatIListKind.1(x₁)) = 0
POL(isNatKind.0(x₁)) = x₁
POL(isNatKind.1(x₁)) = 0
POL(isNatList.0(x₁)) = x₁
POL(isNatList.1(x₁)) = 0
POL(length.0(x₁)) = x₁
POL(length.1(x₁)) = 0
POL(n__0.) = 0
POL(n__and.0-0(x₁, x₂)) = x₂
POL(n__and.0-1(x₁, x₂)) = x₁ + x₂
POL(n__and.1-0(x₁, x₂)) = 0
POL(n__and.1-1(x₁, x₂)) = 1
POL(n__cons.0-0(x₁, x₂)) = x₁ + x₂
POL(n__cons.0-1(x₁, x₂)) = 0
POL(n__cons.1-0(x₁, x₂)) = x₁ + x₂
POL(n__cons.1-1(x₁, x₂)) = 0
POL(n__isNat.0(x₁)) = x₁
POL(n__isNat.1(x₁)) = 0
POL(n__isNatIListKind.0(x₁)) = x₁
POL(n__isNatIListKind.1(x₁)) = 0
POL(n__isNatKind.0(x₁)) = x₁
POL(n__isNatKind.1(x₁)) = 0
POL(n__length.0(x₁)) = x₁
POL(n__length.1(x₁)) = 0
POL(n__nil.) = 0
POL(n__s.0(x₁)) = x₁
POL(n__s.1(x₁)) = x₁
POL(n__zeros.) = 0
POL(nil.) = 0
POL(s.0(x₁)) = x₁
POL(s.1(x₁)) = x₁
POL(tt.) = 0
POL(zeros.) = 0

The following usable rules [17] were oriented:

length.1(X) → n__length.1(X)
cons.1-1(X1, X2) → n__cons.1-1(X1, X2)
and.1-0(X1, X2) → n__and.1-0(X1, X2)
activate.0(X) → X
U61.0-0(tt., L) → s.0(length.0(activate.0(L)))
U22.0(tt.) → tt.
isNatIListKind.1(X) → n__isNatIListKind.1(X)
U51.0-0-0(tt., V1, V2) → U52.0-0(isNat.0(activate.0(V1)), activate.0(V2))
isNatIListKind.0(n__zeros.) → tt.
activate.1(n__nil.) → nil.
activate.0(n__cons.1-1(X1, X2)) → cons.1-1(activate.1(X1), X2)
activate.0(n__cons.0-0(X1, X2)) → cons.0-0(activate.0(X1), X2)
isNatIListKind.0(X) → n__isNatIListKind.0(X)
isNat.0(n__length.0(V1)) → U11.0-0(isNatIListKind.0(activate.0(V1)), activate.0(V1))
s.1(X) → n__s.1(X)
isNat.0(n__s.1(V1)) → U21.0-1(isNatKind.1(activate.1(V1)), activate.1(V1))
zeros. → n__zeros.
U11.0-0(tt., V1) → U12.0(isNatList.0(activate.0(V1)))
U11.0-1(tt., V1) → U12.0(isNatList.1(activate.1(V1)))
activate.0(n__isNatKind.1(X)) → isNatKind.1(X)
activate.1(n__and.1-1(X1, X2)) → and.1-1(activate.1(X1), X2)
isNat.1(X) → n__isNat.1(X)
activate.0(n__isNatIListKind.1(X)) → isNatIListKind.1(X)
nil. → n__nil.
U51.0-1-1(tt., V1, V2) → U52.0-1(isNat.1(activate.1(V1)), activate.1(V2))
activate.0(n__length.0(X)) → length.0(activate.0(X))
and.0-1(tt., X) → activate.1(X)
activate.1(n__and.0-1(X1, X2)) → and.0-1(activate.0(X1), X2)
isNatKind.0(n__s.1(V1)) → isNatKind.1(activate.1(V1))
isNatList.0(n__cons.0-0(V1, V2)) → U51.0-0-0(and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.0(activate.0(V2))), activate.0(V1), activate.0(V2))
U52.0-0(tt., V2) → U53.0(isNatList.0(activate.0(V2)))
U51.0-1-0(tt., V1, V2) → U52.0-0(isNat.1(activate.1(V1)), activate.0(V2))
activate.1(n__length.1(X)) → length.1(activate.1(X))
activate.0(n__cons.1-0(X1, X2)) → cons.1-0(activate.1(X1), X2)
isNatList.0(n__cons.1-0(V1, V2)) → U51.0-1-0(and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.0(activate.0(V2))), activate.1(V1), activate.0(V2))
isNatKind.1(n__0.) → tt.
isNatIListKind.1(n__nil.) → tt.
activate.1(X) → X
activate.0(n__isNat.0(X)) → isNat.0(X)
isNatKind.1(X) → n__isNatKind.1(X)
length.0(cons.1-0(N, L)) → U61.0-0(and.0-0(and.0-0(isNatList.0(activate.0(L)), n__isNatIListKind.0(activate.0(L))), n__and.0-0(n__isNat.1(N), n__isNatKind.1(N))), activate.0(L))
0. → n__0.
isNatKind.1(n__length.1(V1)) → isNatIListKind.1(activate.1(V1))
and.0-0(X1, X2) → n__and.0-0(X1, X2)
activate.0(n__s.0(X)) → s.0(activate.0(X))
isNat.0(n__s.0(V1)) → U21.0-0(isNatKind.0(activate.0(V1)), activate.0(V1))
activate.0(n__zeros.) → zeros.
length.0(cons.0-1(N, L)) → U61.0-1(and.0-0(and.0-0(isNatList.1(activate.1(L)), n__isNatIListKind.1(activate.1(L))), n__and.0-0(n__isNat.0(N), n__isNatKind.0(N))), activate.1(L))
U53.0(tt.) → tt.
activate.0(n__and.1-0(X1, X2)) → and.1-0(activate.1(X1), X2)
activate.1(n__0.) → 0.
U12.0(tt.) → tt.
U61.0-1(tt., L) → s.1(length.1(activate.1(L)))
and.0-1(X1, X2) → n__and.0-1(X1, X2)
U21.0-0(tt., V1) → U22.0(isNat.0(activate.0(V1)))
length.0(cons.1-1(N, L)) → U61.0-1(and.0-0(and.0-0(isNatList.1(activate.1(L)), n__isNatIListKind.1(activate.1(L))), n__and.0-0(n__isNat.1(N), n__isNatKind.1(N))), activate.1(L))
isNat.1(n__0.) → tt.
activate.0(n__cons.0-1(X1, X2)) → cons.0-1(activate.0(X1), X2)
isNat.1(n__length.1(V1)) → U11.0-1(isNatIListKind.1(activate.1(V1)), activate.1(V1))
isNatKind.0(X) → n__isNatKind.0(X)
s.0(X) → n__s.0(X)
cons.0-1(X1, X2) → n__cons.0-1(X1, X2)
cons.0-0(X1, X2) → n__cons.0-0(X1, X2)
activate.0(n__isNat.1(X)) → isNat.1(X)
zeros. → cons.1-0(0., n__zeros.)
length.0(cons.0-0(N, L)) → U61.0-0(and.0-0(and.0-0(isNatList.0(activate.0(L)), n__isNatIListKind.0(activate.0(L))), n__and.0-0(n__isNat.0(N), n__isNatKind.0(N))), activate.0(L))
isNat.0(X) → n__isNat.0(X)
activate.0(n__s.1(X)) → s.1(activate.1(X))
isNatIListKind.0(n__cons.1-0(V1, V2)) → and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.0(activate.0(V2)))
isNatIListKind.0(n__cons.1-1(V1, V2)) → and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.1(activate.1(V2)))
isNatIListKind.0(n__cons.0-0(V1, V2)) → and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.0(activate.0(V2)))
isNatIListKind.0(n__cons.0-1(V1, V2)) → and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.1(activate.1(V2)))
and.0-0(tt., X) → activate.0(X)
activate.0(n__and.0-0(X1, X2)) → and.0-0(activate.0(X1), X2)
isNatKind.0(n__s.0(V1)) → isNatKind.0(activate.0(V1))
isNatKind.0(n__length.0(V1)) → isNatIListKind.0(activate.0(V1))
activate.0(n__isNatIListKind.0(X)) → isNatIListKind.0(X)
activate.0(n__isNatKind.0(X)) → isNatKind.0(X)
and.1-1(X1, X2) → n__and.1-1(X1, X2)
length.1(nil.) → 0.
cons.1-0(X1, X2) → n__cons.1-0(X1, X2)
U52.0-1(tt., V2) → U53.0(isNatList.1(activate.1(V2)))
isNatList.0(n__cons.0-1(V1, V2)) → U51.0-0-1(and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.1(activate.1(V2))), activate.0(V1), activate.1(V2))
length.0(X) → n__length.0(X)
isNatList.0(n__cons.1-1(V1, V2)) → U51.0-1-1(and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.1(activate.1(V2))), activate.1(V1), activate.1(V2))
U21.0-1(tt., V1) → U22.0(isNat.1(activate.1(V1)))
U51.0-0-1(tt., V1, V2) → U52.0-1(isNat.0(activate.0(V1)), activate.1(V2))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ 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

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ SemLabProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                              ↳ AND

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ QDPOrderProof

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ QDPOrderProof

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                            ↳ SemLabProof2

                              ↳ QDP

                    ↳ QDP

          ↳ QDP

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

AND.0-0(tt., X) → ACTIVATE.0(X)
ISNATILISTKIND.0(n__cons.1-0(x0, y1)) → AND.0-0(isNatKind.1(x0), n__isNatIListKind.0(activate.0(y1)))
ISNATILISTKIND.0(n__cons.1-0(n__0., y1)) → AND.0-0(isNatKind.1(0.), n__isNatIListKind.0(activate.0(y1)))
ACTIVATE.0(n__isNatIListKind.0(X)) → ISNATILISTKIND.0(X)

The TRS R consists of the following rules:

U12.0(tt.) → tt.
U61.0-1(tt., L) → s.1(length.1(activate.1(L)))
length.0(cons.0-1(N, L)) → U61.0-1(and.0-0(and.0-0(isNatList.1(activate.1(L)), n__isNatIListKind.1(activate.1(L))), n__and.0-0(n__isNat.0(N), n__isNatKind.0(N))), activate.1(L))
U53.0(tt.) → tt.
activate.0(n__and.1-0(X1, X2)) → and.1-0(activate.1(X1), X2)
activate.1(n__0.) → 0.
isNatIList.0(n__cons.0-1(V1, V2)) → U41.0-0-1(and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.1(activate.1(V2))), activate.0(V1), activate.1(V2))
isNatIList.1(V) → U31.0-1(isNatIListKind.1(activate.1(V)), activate.1(V))
activate.0(n__zeros.) → zeros.
and.0-1(tt., X) → activate.1(X)
isNatKind.1(n__length.1(V1)) → isNatIListKind.1(activate.1(V1))
and.0-0(X1, X2) → n__and.0-0(X1, X2)
activate.0(n__s.0(X)) → s.0(activate.0(X))
isNat.0(n__s.0(V1)) → U21.0-0(isNatKind.0(activate.0(V1)), activate.0(V1))
U51.0-0-1(tt., V1, V2) → U52.0-1(isNat.0(activate.0(V1)), activate.1(V2))
U21.0-1(tt., V1) → U22.0(isNat.1(activate.1(V1)))
isNatList.0(n__cons.1-1(V1, V2)) → U51.0-1-1(and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.1(activate.1(V2))), activate.1(V1), activate.1(V2))
length.0(X) → n__length.0(X)
isNatList.0(n__cons.0-1(V1, V2)) → U51.0-0-1(and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.1(activate.1(V2))), activate.0(V1), activate.1(V2))
U52.0-1(tt., V2) → U53.0(isNatList.1(activate.1(V2)))
cons.1-0(X1, X2) → n__cons.1-0(X1, X2)
length.1(nil.) → 0.
isNatIList.0(n__cons.1-1(V1, V2)) → U41.0-1-1(and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.1(activate.1(V2))), activate.1(V1), activate.1(V2))
and.1-1(X1, X2) → n__and.1-1(X1, X2)
activate.0(n__isNatKind.0(X)) → isNatKind.0(X)
activate.0(n__s.1(X)) → s.1(activate.1(X))
U31.0-0(tt., V) → U32.0(isNatList.0(activate.0(V)))
isNatIListKind.0(n__cons.0-0(V1, V2)) → and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.0(activate.0(V2)))
isNat.0(X) → n__isNat.0(X)
length.0(cons.0-0(N, L)) → U61.0-0(and.0-0(and.0-0(isNatList.0(activate.0(L)), n__isNatIListKind.0(activate.0(L))), n__and.0-0(n__isNat.0(N), n__isNatKind.0(N))), activate.0(L))
activate.0(n__and.0-0(X1, X2)) → and.0-0(activate.0(X1), X2)
zeros. → cons.1-0(0., n__zeros.)
activate.0(n__isNat.1(X)) → isNat.1(X)
activate.0(n__isNatIListKind.0(X)) → isNatIListKind.0(X)
cons.0-0(X1, X2) → n__cons.0-0(X1, X2)
isNatIListKind.0(n__cons.1-1(V1, V2)) → and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.1(activate.1(V2)))
s.0(X) → n__s.0(X)
cons.0-1(X1, X2) → n__cons.0-1(X1, X2)
U41.0-1-0(tt., V1, V2) → U42.0-0(isNat.1(activate.1(V1)), activate.0(V2))
isNatKind.0(X) → n__isNatKind.0(X)
activate.0(n__cons.0-1(X1, X2)) → cons.0-1(activate.0(X1), X2)
isNat.1(n__length.1(V1)) → U11.0-1(isNatIListKind.1(activate.1(V1)), activate.1(V1))
length.0(cons.1-1(N, L)) → U61.0-1(and.0-0(and.0-0(isNatList.1(activate.1(L)), n__isNatIListKind.1(activate.1(L))), n__and.0-0(n__isNat.1(N), n__isNatKind.1(N))), activate.1(L))
isNat.1(n__0.) → tt.
and.0-1(X1, X2) → n__and.0-1(X1, X2)
U21.0-0(tt., V1) → U22.0(isNat.0(activate.0(V1)))
isNatKind.0(n__length.0(V1)) → isNatIListKind.0(activate.0(V1))
zeros. → n__zeros.
U11.0-1(tt., V1) → U12.0(isNatList.1(activate.1(V1)))
U11.0-0(tt., V1) → U12.0(isNatList.0(activate.0(V1)))
U43.0(tt.) → tt.
activate.0(n__isNatKind.1(X)) → isNatKind.1(X)
isNat.1(X) → n__isNat.1(X)
activate.1(n__and.1-1(X1, X2)) → and.1-1(activate.1(X1), X2)
U31.0-1(tt., V) → U32.0(isNatList.1(activate.1(V)))
activate.0(n__cons.0-0(X1, X2)) → cons.0-0(activate.0(X1), X2)
U41.0-0-0(tt., V1, V2) → U42.0-0(isNat.0(activate.0(V1)), activate.0(V2))
U32.0(tt.) → tt.
isNat.0(n__length.0(V1)) → U11.0-0(isNatIListKind.0(activate.0(V1)), activate.0(V1))
isNatIListKind.0(X) → n__isNatIListKind.0(X)
isNat.0(n__s.1(V1)) → U21.0-1(isNatKind.1(activate.1(V1)), activate.1(V1))
s.1(X) → n__s.1(X)
U22.0(tt.) → tt.
isNatIListKind.1(X) → n__isNatIListKind.1(X)
U51.0-0-0(tt., V1, V2) → U52.0-0(isNat.0(activate.0(V1)), activate.0(V2))
isNatIListKind.0(n__zeros.) → tt.
activate.1(n__nil.) → nil.
isNatKind.0(n__s.0(V1)) → isNatKind.0(activate.0(V1))
isNatIList.0(n__cons.0-0(V1, V2)) → U41.0-0-0(and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.0(activate.0(V2))), activate.0(V1), activate.0(V2))
activate.0(n__cons.1-1(X1, X2)) → cons.1-1(activate.1(X1), X2)
isNatIList.0(n__cons.1-0(V1, V2)) → U41.0-1-0(and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.0(activate.0(V2))), activate.1(V1), activate.0(V2))
and.0-0(tt., X) → activate.0(X)
U42.0-1(tt., V2) → U43.0(isNatIList.1(activate.1(V2)))
length.1(X) → n__length.1(X)
cons.1-1(X1, X2) → n__cons.1-1(X1, X2)
and.1-0(X1, X2) → n__and.1-0(X1, X2)
activate.0(X) → X
U61.0-0(tt., L) → s.0(length.0(activate.0(L)))
activate.0(n__isNat.0(X)) → isNat.0(X)
activate.1(X) → X
isNatIListKind.1(n__nil.) → tt.
isNatKind.1(n__0.) → tt.
0. → n__0.
U41.0-1-1(tt., V1, V2) → U42.0-1(isNat.1(activate.1(V1)), activate.1(V2))
length.0(cons.1-0(N, L)) → U61.0-0(and.0-0(and.0-0(isNatList.0(activate.0(L)), n__isNatIListKind.0(activate.0(L))), n__and.0-0(n__isNat.1(N), n__isNatKind.1(N))), activate.0(L))
isNatKind.1(X) → n__isNatKind.1(X)
isNatIListKind.0(n__cons.1-0(V1, V2)) → and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.0(activate.0(V2)))
isNatIListKind.0(n__cons.0-1(V1, V2)) → and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.1(activate.1(V2)))
isNatIList.0(V) → U31.0-0(isNatIListKind.0(activate.0(V)), activate.0(V))
U42.0-0(tt., V2) → U43.0(isNatIList.0(activate.0(V2)))
isNatList.0(n__cons.1-0(V1, V2)) → U51.0-1-0(and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.0(activate.0(V2))), activate.1(V1), activate.0(V2))
activate.0(n__cons.1-0(X1, X2)) → cons.1-0(activate.1(X1), X2)
activate.1(n__length.1(X)) → length.1(activate.1(X))
isNatIList.0(n__zeros.) → tt.
activate.0(n__length.0(X)) → length.0(activate.0(X))
activate.1(n__and.0-1(X1, X2)) → and.0-1(activate.0(X1), X2)
U51.0-1-1(tt., V1, V2) → U52.0-1(isNat.1(activate.1(V1)), activate.1(V2))
U41.0-0-1(tt., V1, V2) → U42.0-1(isNat.0(activate.0(V1)), activate.1(V2))
U52.0-0(tt., V2) → U53.0(isNatList.0(activate.0(V2)))
U51.0-1-0(tt., V1, V2) → U52.0-0(isNat.1(activate.1(V1)), activate.0(V2))
isNatKind.0(n__s.1(V1)) → isNatKind.1(activate.1(V1))
isNatList.0(n__cons.0-0(V1, V2)) → U51.0-0-0(and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.0(activate.0(V2))), activate.0(V1), activate.0(V2))
nil. → n__nil.
isNatList.1(n__nil.) → tt.
activate.0(n__isNatIListKind.1(X)) → isNatIListKind.1(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

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ 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

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ SemLabProof

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                                                                                              ↳ AND

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                            ↳ SemLabProof2

                              ↳ QDP

                    ↳ QDP

          ↳ QDP

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

ISNATKIND.0(n__s.0(n__and.1-0(x0, x1))) → ISNATKIND.0(and.1-0(activate.1(x0), x1))
ISNATKIND.0(n__s.0(n__s.0(x0))) → ISNATKIND.0(s.0(activate.0(x0)))
ISNATKIND.0(n__s.0(n__length.0(x0))) → ISNATKIND.0(length.0(activate.0(x0)))
ISNATKIND.0(n__s.0(x0)) → ISNATKIND.0(x0)
ISNATKIND.0(n__s.0(n__and.0-0(x0, x1))) → ISNATKIND.0(and.0-0(activate.0(x0), x1))
ISNATKIND.0(n__s.0(n__s.1(x0))) → ISNATKIND.0(s.1(activate.1(x0)))

The TRS R consists of the following rules:

U12.0(tt.) → tt.
U61.0-1(tt., L) → s.1(length.1(activate.1(L)))
length.0(cons.0-1(N, L)) → U61.0-1(and.0-0(and.0-0(isNatList.1(activate.1(L)), n__isNatIListKind.1(activate.1(L))), n__and.0-0(n__isNat.0(N), n__isNatKind.0(N))), activate.1(L))
U53.0(tt.) → tt.
activate.0(n__and.1-0(X1, X2)) → and.1-0(activate.1(X1), X2)
activate.1(n__0.) → 0.
isNatIList.0(n__cons.0-1(V1, V2)) → U41.0-0-1(and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.1(activate.1(V2))), activate.0(V1), activate.1(V2))
isNatIList.1(V) → U31.0-1(isNatIListKind.1(activate.1(V)), activate.1(V))
activate.0(n__zeros.) → zeros.
and.0-1(tt., X) → activate.1(X)
isNatKind.1(n__length.1(V1)) → isNatIListKind.1(activate.1(V1))
and.0-0(X1, X2) → n__and.0-0(X1, X2)
activate.0(n__s.0(X)) → s.0(activate.0(X))
isNat.0(n__s.0(V1)) → U21.0-0(isNatKind.0(activate.0(V1)), activate.0(V1))
U51.0-0-1(tt., V1, V2) → U52.0-1(isNat.0(activate.0(V1)), activate.1(V2))
U21.0-1(tt., V1) → U22.0(isNat.1(activate.1(V1)))
isNatList.0(n__cons.1-1(V1, V2)) → U51.0-1-1(and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.1(activate.1(V2))), activate.1(V1), activate.1(V2))
length.0(X) → n__length.0(X)
isNatList.0(n__cons.0-1(V1, V2)) → U51.0-0-1(and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.1(activate.1(V2))), activate.0(V1), activate.1(V2))
U52.0-1(tt., V2) → U53.0(isNatList.1(activate.1(V2)))
cons.1-0(X1, X2) → n__cons.1-0(X1, X2)
length.1(nil.) → 0.
isNatIList.0(n__cons.1-1(V1, V2)) → U41.0-1-1(and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.1(activate.1(V2))), activate.1(V1), activate.1(V2))
and.1-1(X1, X2) → n__and.1-1(X1, X2)
activate.0(n__isNatKind.0(X)) → isNatKind.0(X)
activate.0(n__s.1(X)) → s.1(activate.1(X))
U31.0-0(tt., V) → U32.0(isNatList.0(activate.0(V)))
isNatIListKind.0(n__cons.0-0(V1, V2)) → and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.0(activate.0(V2)))
isNat.0(X) → n__isNat.0(X)
length.0(cons.0-0(N, L)) → U61.0-0(and.0-0(and.0-0(isNatList.0(activate.0(L)), n__isNatIListKind.0(activate.0(L))), n__and.0-0(n__isNat.0(N), n__isNatKind.0(N))), activate.0(L))
activate.0(n__and.0-0(X1, X2)) → and.0-0(activate.0(X1), X2)
zeros. → cons.1-0(0., n__zeros.)
activate.0(n__isNat.1(X)) → isNat.1(X)
activate.0(n__isNatIListKind.0(X)) → isNatIListKind.0(X)
cons.0-0(X1, X2) → n__cons.0-0(X1, X2)
isNatIListKind.0(n__cons.1-1(V1, V2)) → and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.1(activate.1(V2)))
s.0(X) → n__s.0(X)
cons.0-1(X1, X2) → n__cons.0-1(X1, X2)
U41.0-1-0(tt., V1, V2) → U42.0-0(isNat.1(activate.1(V1)), activate.0(V2))
isNatKind.0(X) → n__isNatKind.0(X)
activate.0(n__cons.0-1(X1, X2)) → cons.0-1(activate.0(X1), X2)
isNat.1(n__length.1(V1)) → U11.0-1(isNatIListKind.1(activate.1(V1)), activate.1(V1))
length.0(cons.1-1(N, L)) → U61.0-1(and.0-0(and.0-0(isNatList.1(activate.1(L)), n__isNatIListKind.1(activate.1(L))), n__and.0-0(n__isNat.1(N), n__isNatKind.1(N))), activate.1(L))
isNat.1(n__0.) → tt.
and.0-1(X1, X2) → n__and.0-1(X1, X2)
U21.0-0(tt., V1) → U22.0(isNat.0(activate.0(V1)))
isNatKind.0(n__length.0(V1)) → isNatIListKind.0(activate.0(V1))
zeros. → n__zeros.
U11.0-1(tt., V1) → U12.0(isNatList.1(activate.1(V1)))
U11.0-0(tt., V1) → U12.0(isNatList.0(activate.0(V1)))
U43.0(tt.) → tt.
activate.0(n__isNatKind.1(X)) → isNatKind.1(X)
isNat.1(X) → n__isNat.1(X)
activate.1(n__and.1-1(X1, X2)) → and.1-1(activate.1(X1), X2)
U31.0-1(tt., V) → U32.0(isNatList.1(activate.1(V)))
activate.0(n__cons.0-0(X1, X2)) → cons.0-0(activate.0(X1), X2)
U41.0-0-0(tt., V1, V2) → U42.0-0(isNat.0(activate.0(V1)), activate.0(V2))
U32.0(tt.) → tt.
isNat.0(n__length.0(V1)) → U11.0-0(isNatIListKind.0(activate.0(V1)), activate.0(V1))
isNatIListKind.0(X) → n__isNatIListKind.0(X)
isNat.0(n__s.1(V1)) → U21.0-1(isNatKind.1(activate.1(V1)), activate.1(V1))
s.1(X) → n__s.1(X)
U22.0(tt.) → tt.
isNatIListKind.1(X) → n__isNatIListKind.1(X)
U51.0-0-0(tt., V1, V2) → U52.0-0(isNat.0(activate.0(V1)), activate.0(V2))
isNatIListKind.0(n__zeros.) → tt.
activate.1(n__nil.) → nil.
isNatKind.0(n__s.0(V1)) → isNatKind.0(activate.0(V1))
isNatIList.0(n__cons.0-0(V1, V2)) → U41.0-0-0(and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.0(activate.0(V2))), activate.0(V1), activate.0(V2))
activate.0(n__cons.1-1(X1, X2)) → cons.1-1(activate.1(X1), X2)
isNatIList.0(n__cons.1-0(V1, V2)) → U41.0-1-0(and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.0(activate.0(V2))), activate.1(V1), activate.0(V2))
and.0-0(tt., X) → activate.0(X)
U42.0-1(tt., V2) → U43.0(isNatIList.1(activate.1(V2)))
length.1(X) → n__length.1(X)
cons.1-1(X1, X2) → n__cons.1-1(X1, X2)
and.1-0(X1, X2) → n__and.1-0(X1, X2)
activate.0(X) → X
U61.0-0(tt., L) → s.0(length.0(activate.0(L)))
activate.0(n__isNat.0(X)) → isNat.0(X)
activate.1(X) → X
isNatIListKind.1(n__nil.) → tt.
isNatKind.1(n__0.) → tt.
0. → n__0.
U41.0-1-1(tt., V1, V2) → U42.0-1(isNat.1(activate.1(V1)), activate.1(V2))
length.0(cons.1-0(N, L)) → U61.0-0(and.0-0(and.0-0(isNatList.0(activate.0(L)), n__isNatIListKind.0(activate.0(L))), n__and.0-0(n__isNat.1(N), n__isNatKind.1(N))), activate.0(L))
isNatKind.1(X) → n__isNatKind.1(X)
isNatIListKind.0(n__cons.1-0(V1, V2)) → and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.0(activate.0(V2)))
isNatIListKind.0(n__cons.0-1(V1, V2)) → and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.1(activate.1(V2)))
isNatIList.0(V) → U31.0-0(isNatIListKind.0(activate.0(V)), activate.0(V))
U42.0-0(tt., V2) → U43.0(isNatIList.0(activate.0(V2)))
isNatList.0(n__cons.1-0(V1, V2)) → U51.0-1-0(and.0-0(isNatKind.1(activate.1(V1)), n__isNatIListKind.0(activate.0(V2))), activate.1(V1), activate.0(V2))
activate.0(n__cons.1-0(X1, X2)) → cons.1-0(activate.1(X1), X2)
activate.1(n__length.1(X)) → length.1(activate.1(X))
isNatIList.0(n__zeros.) → tt.
activate.0(n__length.0(X)) → length.0(activate.0(X))
activate.1(n__and.0-1(X1, X2)) → and.0-1(activate.0(X1), X2)
U51.0-1-1(tt., V1, V2) → U52.0-1(isNat.1(activate.1(V1)), activate.1(V2))
U41.0-0-1(tt., V1, V2) → U42.0-1(isNat.0(activate.0(V1)), activate.1(V2))
U52.0-0(tt., V2) → U53.0(isNatList.0(activate.0(V2)))
U51.0-1-0(tt., V1, V2) → U52.0-0(isNat.1(activate.1(V1)), activate.0(V2))
isNatKind.0(n__s.1(V1)) → isNatKind.1(activate.1(V1))
isNatList.0(n__cons.0-0(V1, V2)) → U51.0-0-0(and.0-0(isNatKind.0(activate.0(V1)), n__isNatIListKind.0(activate.0(V2))), activate.0(V1), activate.0(V2))
nil. → n__nil.
isNatList.1(n__nil.) → tt.
activate.0(n__isNatIListKind.1(X)) → isNatIListKind.1(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
As can be seen after transforming the QDP problem by semantic labelling [33] and then some rule deleting processors, only certain labelled rules and pairs can be used. Hence, we only have to consider all unlabelled pairs and rules (without the decreasing rules for quasi-models).

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ 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

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ SemLabProof

                                                                                                                                                                                            ↳ SemLabProof2

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ DependencyGraphProof

                              ↳ QDP

                    ↳ QDP

          ↳ QDP

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

ISNATILISTKIND(n__cons(x0, y1)) → AND(isNatKind(x0), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(x0)) → ISNATKIND(x0)
ISNATKIND(n__s(n__length(x0))) → ISNATKIND(length(activate(x0)))
ISNATKIND(n__s(n__s(x0))) → ISNATKIND(s(activate(x0)))
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
AND(tt, X) → ACTIVATE(X)
ISNATILISTKIND(n__cons(n__0, y1)) → AND(isNatKind(0), n__isNatIListKind(activate(y1)))
ISNATKIND(n__s(n__and(x0, x1))) → ISNATKIND(and(activate(x0), x1))

The TRS R consists of the following rules:

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ 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

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ SemLabProof

                                                                                                                                                                                            ↳ SemLabProof2

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                  ↳ AND

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                    ↳ QDP

                              ↳ QDP

                    ↳ QDP

          ↳ QDP

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

ISNATKIND(n__s(n__s(x0))) → ISNATKIND(s(activate(x0)))
ISNATKIND(n__s(n__length(x0))) → ISNATKIND(length(activate(x0)))
ISNATKIND(n__s(x0)) → ISNATKIND(x0)
ISNATKIND(n__s(n__and(x0, x1))) → ISNATKIND(and(activate(x0), x1))

The TRS R consists of the following rules:

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ 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

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                                      ↳ QDP

                                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                                          ↳ QDP

                                                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                                                  ↳ QDP

                                                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                                                      ↳ QDP

                                                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                                                          ↳ QDP

                                                                                                                                                                                            ↳ SemLabProof

                                                                                                                                                                                            ↳ SemLabProof2

                                                                                                                                                                                              ↳ QDP

                                                                                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                                                                                  ↳ AND

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                    ↳ QDP

                              ↳ QDP

                    ↳ QDP

          ↳ QDP

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

ISNATILISTKIND(n__cons(x0, y1)) → AND(isNatKind(x0), n__isNatIListKind(activate(y1)))
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
AND(tt, X) → ACTIVATE(X)
ISNATILISTKIND(n__cons(n__0, y1)) → AND(isNatKind(0), n__isNatIListKind(activate(y1)))

The TRS R consists of the following rules:

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                              ↳ QDP

                    ↳ QDP

          ↳ QDP

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

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

The TRS R consists of the following rules:

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ AND

                    ↳ QDP

                    ↳ QDP

                    ↳ QDP

          ↳ QDP

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

ISNATLIST(n__cons(V1, V2)) → U51¹(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U51¹(tt, V1, V2) → U52¹(isNat(activate(V1)), activate(V2))
U52¹(tt, V2) → ISNATLIST(activate(V2))

The TRS R consists of the following rules:

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

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

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

The TRS R consists of the following rules:

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

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