Termination w.r.t. Q proof of /tmp/tmpKY6udV/LengthOfFiniteLists_nokinds-noand

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

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

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

Q is empty.

↳ QTRS

  ↳ RRRPoloQTRSProof

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

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

Q is empty.

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

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
length(nil) → 0
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

Q is empty.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:

isNatList(n__nil) → tt
length(nil) → 0

Used ordering:
Polynomial interpretation [25]:

POL(0) = 0
POL(U11(x₁)) = x₁
POL(U21(x₁)) = x₁
POL(U31(x₁)) = x₁
POL(U41(x₁, x₂)) = 2·x₁ + 2·x₂
POL(U42(x₁)) = x₁
POL(U51(x₁, x₂)) = 2·x₁ + 2·x₂
POL(U52(x₁)) = x₁
POL(U61(x₁, x₂, x₃)) = x₁ + 2·x₂ + 2·x₃
POL(U62(x₁, x₂)) = 2·x₁ + 2·x₂
POL(activate(x₁)) = x₁
POL(cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(isNat(x₁)) = x₁
POL(isNatIList(x₁)) = 2·x₁
POL(isNatList(x₁)) = 2·x₁
POL(length(x₁)) = 2·x₁
POL(n__0) = 0
POL(n__cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(n__length(x₁)) = 2·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

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

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

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

Q is empty.

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

zeros → cons(0, n__zeros)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

Q is empty.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:

U11(tt) → tt
isNat(n__length(V1)) → U11(isNatList(activate(V1)))

Used ordering:
Polynomial interpretation [25]:

POL(0) = 0
POL(U11(x₁)) = 1 + x₁
POL(U21(x₁)) = x₁
POL(U31(x₁)) = x₁
POL(U41(x₁, x₂)) = 2·x₁ + 2·x₂
POL(U42(x₁)) = x₁
POL(U51(x₁, x₂)) = 2·x₁ + 2·x₂
POL(U52(x₁)) = x₁
POL(U61(x₁, x₂, x₃)) = 1 + x₁ + 2·x₂ + 2·x₃
POL(U62(x₁, x₂)) = 1 + x₁ + 2·x₂
POL(activate(x₁)) = x₁
POL(cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(isNat(x₁)) = 2·x₁
POL(isNatIList(x₁)) = 2·x₁
POL(isNatList(x₁)) = 2·x₁
POL(length(x₁)) = 1 + 2·x₁
POL(n__0) = 0
POL(n__cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(n__length(x₁)) = 1 + 2·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

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

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

zeros → cons(0, n__zeros)
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

Q is empty.

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

zeros → cons(0, n__zeros)
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

Q is empty.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:

isNatIList(V) → U31(isNatList(activate(V)))
isNatIList(n__zeros) → tt

Used ordering:
Polynomial interpretation [25]:

POL(0) = 0
POL(U21(x₁)) = x₁
POL(U31(x₁)) = x₁
POL(U41(x₁, x₂)) = 1 + 2·x₁ + 2·x₂
POL(U42(x₁)) = x₁
POL(U51(x₁, x₂)) = x₁ + 2·x₂
POL(U52(x₁)) = x₁
POL(U61(x₁, x₂, x₃)) = x₁ + 2·x₂ + x₃
POL(U62(x₁, x₂)) = x₁ + 2·x₂
POL(activate(x₁)) = x₁
POL(cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(isNat(x₁)) = x₁
POL(isNatIList(x₁)) = 1 + 2·x₁
POL(isNatList(x₁)) = 2·x₁
POL(length(x₁)) = 2·x₁
POL(n__0) = 0
POL(n__cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(n__length(x₁)) = 2·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

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

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

zeros → cons(0, n__zeros)
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

Q is empty.

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

zeros → cons(0, n__zeros)
U21(tt) → tt
U31(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

Q is empty.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:

U31(tt) → tt

Used ordering:
Polynomial interpretation [25]:

POL(0) = 0
POL(U21(x₁)) = x₁
POL(U31(x₁)) = 1 + 2·x₁
POL(U41(x₁, x₂)) = 2·x₁ + 2·x₂
POL(U42(x₁)) = 2·x₁
POL(U51(x₁, x₂)) = x₁ + 2·x₂
POL(U52(x₁)) = x₁
POL(U61(x₁, x₂, x₃)) = 2·x₁ + 2·x₂ + 2·x₃
POL(U62(x₁, x₂)) = 2·x₁ + 2·x₂
POL(activate(x₁)) = x₁
POL(cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(isNat(x₁)) = x₁
POL(isNatIList(x₁)) = x₁
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = 2·x₁
POL(n__0) = 0
POL(n__cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(n__length(x₁)) = 2·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

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

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

zeros → cons(0, n__zeros)
U21(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

Q is empty.

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

ISNATILIST(n__cons(V1, V2)) → ISNAT(activate(V1))
U61¹(tt, L, N) → ISNAT(activate(N))
U51¹(tt, V2) → ISNATLIST(activate(V2))
U51¹(tt, V2) → U52¹(isNatList(activate(V2)))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
U51¹(tt, V2) → ACTIVATE(V2)
U61¹(tt, L, N) → U62¹(isNat(activate(N)), activate(L))
ACTIVATE(n__0) → 0¹
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__cons(X1, X2)) → CONS(X1, X2)
U41¹(tt, V2) → ISNATILIST(activate(V2))
U62¹(tt, L) → LENGTH(activate(L))
ISNAT(n__s(V1)) → U21¹(isNat(activate(V1)))
U62¹(tt, L) → S(length(activate(L)))
ACTIVATE(n__zeros) → ZEROS
U41¹(tt, V2) → U42¹(isNatIList(activate(V2)))
U41¹(tt, V2) → ACTIVATE(V2)
ACTIVATE(n__length(X)) → LENGTH(X)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
LENGTH(cons(N, L)) → ACTIVATE(L)
LENGTH(cons(N, L)) → U61¹(isNatList(activate(L)), activate(L), N)
ACTIVATE(n__nil) → NIL
U61¹(tt, L, N) → ACTIVATE(L)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNATILIST(n__cons(V1, V2)) → U41¹(isNat(activate(V1)), activate(V2))
ACTIVATE(n__s(X)) → S(X)
ISNAT(n__s(V1)) → ACTIVATE(V1)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
U61¹(tt, L, N) → ACTIVATE(N)
ZEROS → CONS(0, n__zeros)
U62¹(tt, L) → ACTIVATE(L)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATLIST(n__cons(V1, V2)) → U51¹(isNat(activate(V1)), activate(V2))
ZEROS → 0¹

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U21(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

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

ISNATILIST(n__cons(V1, V2)) → ISNAT(activate(V1))
U61¹(tt, L, N) → ISNAT(activate(N))
U51¹(tt, V2) → ISNATLIST(activate(V2))
U51¹(tt, V2) → U52¹(isNatList(activate(V2)))
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
U51¹(tt, V2) → ACTIVATE(V2)
U61¹(tt, L, N) → U62¹(isNat(activate(N)), activate(L))
ACTIVATE(n__0) → 0¹
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__cons(X1, X2)) → CONS(X1, X2)
U41¹(tt, V2) → ISNATILIST(activate(V2))
U62¹(tt, L) → LENGTH(activate(L))
ISNAT(n__s(V1)) → U21¹(isNat(activate(V1)))
U62¹(tt, L) → S(length(activate(L)))
ACTIVATE(n__zeros) → ZEROS
U41¹(tt, V2) → U42¹(isNatIList(activate(V2)))
U41¹(tt, V2) → ACTIVATE(V2)
ACTIVATE(n__length(X)) → LENGTH(X)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
LENGTH(cons(N, L)) → ACTIVATE(L)
LENGTH(cons(N, L)) → U61¹(isNatList(activate(L)), activate(L), N)
ACTIVATE(n__nil) → NIL
U61¹(tt, L, N) → ACTIVATE(L)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNATILIST(n__cons(V1, V2)) → U41¹(isNat(activate(V1)), activate(V2))
ACTIVATE(n__s(X)) → S(X)
ISNAT(n__s(V1)) → ACTIVATE(V1)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
U61¹(tt, L, N) → ACTIVATE(N)
ZEROS → CONS(0, n__zeros)
U62¹(tt, L) → ACTIVATE(L)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATLIST(n__cons(V1, V2)) → U51¹(isNat(activate(V1)), activate(V2))
ZEROS → 0¹

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U21(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

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

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                          ↳ QDP

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

U61¹(tt, L, N) → ISNAT(activate(N))
ACTIVATE(n__length(X)) → LENGTH(X)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
LENGTH(cons(N, L)) → ACTIVATE(L)
U51¹(tt, V2) → ISNATLIST(activate(V2))
LENGTH(cons(N, L)) → U61¹(isNatList(activate(L)), activate(L), N)
U61¹(tt, L, N) → ACTIVATE(L)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
U51¹(tt, V2) → ACTIVATE(V2)
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
U61¹(tt, L, N) → U62¹(isNat(activate(N)), activate(L))
U61¹(tt, L, N) → ACTIVATE(N)
U62¹(tt, L) → LENGTH(activate(L))
U62¹(tt, L) → ACTIVATE(L)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATLIST(n__cons(V1, V2)) → U51¹(isNat(activate(V1)), activate(V2))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U21(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the usable rules with reduction pair processor [15] with a polynomial ordering [25], all dependency pairs and the corresponding usable rules [17] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

No dependency pairs are removed.

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [25]:

POL(0) = 0
POL(ACTIVATE(x₁)) = x₁
POL(ISNAT(x₁)) = 2·x₁
POL(ISNATLIST(x₁)) = 2·x₁
POL(LENGTH(x₁)) = 2·x₁
POL(U21(x₁)) = x₁
POL(U51(x₁, x₂)) = x₁ + 2·x₂
POL(U51¹(x₁, x₂)) = 2·x₁ + 2·x₂
POL(U52(x₁)) = x₁
POL(U61(x₁, x₂, x₃)) = x₁ + 2·x₂ + 2·x₃
POL(U61¹(x₁, x₂, x₃)) = x₁ + 2·x₂ + 2·x₃
POL(U62(x₁, x₂)) = x₁ + 2·x₂
POL(U62¹(x₁, x₂)) = x₁ + 2·x₂
POL(activate(x₁)) = x₁
POL(cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(isNat(x₁)) = 2·x₁
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = 2·x₁
POL(n__0) = 0
POL(n__cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(n__length(x₁)) = 2·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

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ RuleRemovalProof

                          ↳ QDP

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

U61¹(tt, L, N) → ISNAT(activate(N))
ACTIVATE(n__length(X)) → LENGTH(X)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
LENGTH(cons(N, L)) → ACTIVATE(L)
U51¹(tt, V2) → ISNATLIST(activate(V2))
LENGTH(cons(N, L)) → U61¹(isNatList(activate(L)), activate(L), N)
U61¹(tt, L, N) → ACTIVATE(L)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
U51¹(tt, V2) → ACTIVATE(V2)
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
U61¹(tt, L, N) → U62¹(isNat(activate(N)), activate(L))
U61¹(tt, L, N) → ACTIVATE(N)
U62¹(tt, L) → LENGTH(activate(L))
U62¹(tt, L) → ACTIVATE(L)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATLIST(n__cons(V1, V2)) → U51¹(isNat(activate(V1)), activate(V2))

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the rule removal processor [15] with the following polynomial ordering [25], at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

ACTIVATE(n__length(X)) → LENGTH(X)

Used ordering: POLO with Polynomial interpretation [25]:

POL(0) = 0
POL(ACTIVATE(x₁)) = x₁
POL(ISNAT(x₁)) = x₁
POL(ISNATLIST(x₁)) = x₁
POL(LENGTH(x₁)) = x₁
POL(U21(x₁)) = x₁
POL(U51(x₁, x₂)) = x₁ + x₂
POL(U51¹(x₁, x₂)) = 2·x₁ + x₂
POL(U52(x₁)) = x₁
POL(U61(x₁, x₂, x₃)) = 2 + x₁ + x₂ + x₃
POL(U61¹(x₁, x₂, x₃)) = x₁ + x₂ + 2·x₃
POL(U62(x₁, x₂)) = 2 + x₁ + x₂
POL(U62¹(x₁, x₂)) = 2·x₁ + x₂
POL(activate(x₁)) = x₁
POL(cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(isNat(x₁)) = x₁
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = 2 + x₁
POL(n__0) = 0
POL(n__cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(n__length(x₁)) = 2 + 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

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ RuleRemovalProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                          ↳ QDP

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

U61¹(tt, L, N) → ISNAT(activate(N))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
LENGTH(cons(N, L)) → ACTIVATE(L)
U51¹(tt, V2) → ISNATLIST(activate(V2))
LENGTH(cons(N, L)) → U61¹(isNatList(activate(L)), activate(L), N)
U61¹(tt, L, N) → ACTIVATE(L)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
U51¹(tt, V2) → ACTIVATE(V2)
ISNATLIST(n__cons(V1, V2)) → ISNAT(activate(V1))
U61¹(tt, L, N) → U62¹(isNat(activate(N)), activate(L))
U61¹(tt, L, N) → ACTIVATE(N)
U62¹(tt, L) → LENGTH(activate(L))
U62¹(tt, L) → ACTIVATE(L)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATLIST(n__cons(V1, V2)) → U51¹(isNat(activate(V1)), activate(V2))

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ RuleRemovalProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                          ↳ QDPOrderProof

                                        ↳ QDP

                                        ↳ QDP

                          ↳ QDP

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

ISNAT(n__s(V1)) → ISNAT(activate(V1))

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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__s(V1)) → ISNAT(activate(V1))

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(ISNAT(x₁)) = x₁
POL(U21(x₁)) = 1
POL(U51(x₁, x₂)) = 0
POL(U52(x₁)) = x₁
POL(U61(x₁, x₂, x₃)) = x₁
POL(U62(x₁, x₂)) = 1
POL(activate(x₁)) = x₁
POL(cons(x₁, x₂)) = 0
POL(isNat(x₁)) = x₁
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 0
POL(n__0) = 0
POL(n__cons(x₁, x₂)) = 0
POL(n__length(x₁)) = 0
POL(n__nil) = 0
POL(n__s(x₁)) = 1 + x₁
POL(n__zeros) = 0
POL(nil) = 0
POL(s(x₁)) = 1 + x₁
POL(tt) = 1
POL(zeros) = 0

The following usable rules [17] were oriented:

U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
length(X) → n__length(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
zeros → n__zeros
zeros → cons(0, n__zeros)
0 → n__0
U52(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U21(tt) → tt
U62(tt, L) → s(length(activate(L)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
activate(n__nil) → nil
activate(X) → X
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ RuleRemovalProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                          ↳ QDPOrderProof

                                            ↳ QDP

                                              ↳ PisEmptyProof

                                        ↳ QDP

                                        ↳ QDP

                          ↳ QDP

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

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ RuleRemovalProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                        ↳ QDP

                                          ↳ Narrowing

                                        ↳ QDP

                          ↳ QDP

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

U51¹(tt, V2) → ISNATLIST(activate(V2))
ISNATLIST(n__cons(V1, V2)) → U51¹(isNat(activate(V1)), activate(V2))

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

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

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ RuleRemovalProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                        ↳ QDP

                          ↳ QDP

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

ISNATLIST(n__cons(n__length(x0), y1)) → U51¹(isNat(length(x0)), activate(y1))
ISNATLIST(n__cons(x0, y1)) → U51¹(isNat(x0), activate(y1))
ISNATLIST(n__cons(n__nil, y1)) → U51¹(isNat(nil), activate(y1))
ISNATLIST(n__cons(n__cons(x0, x1), y1)) → U51¹(isNat(cons(x0, x1)), activate(y1))
ISNATLIST(n__cons(n__zeros, y1)) → U51¹(isNat(zeros), activate(y1))
U51¹(tt, V2) → ISNATLIST(activate(V2))
ISNATLIST(n__cons(n__0, y1)) → U51¹(isNat(0), activate(y1))
ISNATLIST(n__cons(n__s(x0), y1)) → U51¹(isNat(s(x0)), activate(y1))

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

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

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ RuleRemovalProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                        ↳ QDP

                          ↳ QDP

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

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

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

ISNATLIST(n__cons(n__cons(x0, x1), y2)) → U51¹(isNat(n__cons(x0, x1)), activate(y2))

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ RuleRemovalProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                        ↳ QDP

                          ↳ QDP

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

ISNATLIST(n__cons(n__length(x0), y1)) → U51¹(isNat(length(x0)), activate(y1))
ISNATLIST(n__cons(n__nil, y1)) → U51¹(isNat(nil), activate(y1))
U51¹(tt, n__cons(x0, x1)) → ISNATLIST(cons(x0, x1))
ISNATLIST(n__cons(n__0, y1)) → U51¹(isNat(0), activate(y1))
ISNATLIST(n__cons(n__s(x0), y1)) → U51¹(isNat(s(x0)), activate(y1))
U51¹(tt, n__nil) → ISNATLIST(nil)
U51¹(tt, n__s(x0)) → ISNATLIST(s(x0))
ISNATLIST(n__cons(x0, y1)) → U51¹(isNat(x0), activate(y1))
U51¹(tt, n__length(x0)) → ISNATLIST(length(x0))
U51¹(tt, n__0) → ISNATLIST(0)
U51¹(tt, n__zeros) → ISNATLIST(zeros)
ISNATLIST(n__cons(n__zeros, y1)) → U51¹(isNat(zeros), activate(y1))
U51¹(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(n__cons(x0, x1), y2)) → U51¹(isNat(n__cons(x0, x1)), activate(y2))

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ RuleRemovalProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                        ↳ QDP

                          ↳ QDP

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

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

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

U51¹(tt, n__nil) → ISNATLIST(n__nil)

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ RuleRemovalProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                        ↳ QDP

                          ↳ QDP

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

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

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ RuleRemovalProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                        ↳ QDP

                          ↳ QDP

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

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

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

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

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ RuleRemovalProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                        ↳ QDP

                          ↳ QDP

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

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

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ RuleRemovalProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                        ↳ QDP

                          ↳ QDP

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

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

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

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

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ RuleRemovalProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                        ↳ QDP

                          ↳ QDP

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

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

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ RuleRemovalProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                        ↳ QDP

                          ↳ QDP

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

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

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

U51¹(tt, n__0) → ISNATLIST(n__0)

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ RuleRemovalProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                        ↳ 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

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

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

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ RuleRemovalProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                        ↳ 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

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

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

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

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

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ RuleRemovalProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                        ↳ 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

                                                                                              ↳ Narrowing

                                        ↳ QDP

                          ↳ QDP

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

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

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

ISNATLIST(n__cons(n__nil, y0)) → U51¹(isNat(n__nil), activate(y0))

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ RuleRemovalProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                        ↳ 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

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                        ↳ QDP

                          ↳ QDP

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

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

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ RuleRemovalProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                        ↳ 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

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                        ↳ QDP

                          ↳ QDP

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

ISNATLIST(n__cons(n__length(x0), y1)) → U51¹(isNat(length(x0)), activate(y1))
ISNATLIST(n__cons(x0, y1)) → U51¹(isNat(x0), activate(y1))
ISNATLIST(n__cons(n__zeros, y1)) → U51¹(isNat(zeros), activate(y1))
U51¹(tt, n__length(x0)) → ISNATLIST(length(x0))
U51¹(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
U51¹(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(n__0, y1)) → U51¹(isNat(0), activate(y1))
ISNATLIST(n__cons(n__s(x0), y1)) → U51¹(isNat(s(x0)), activate(y1))
U51¹(tt, n__zeros) → ISNATLIST(cons(0, n__zeros))

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

ISNATLIST(n__cons(n__zeros, y0)) → U51¹(isNat(cons(0, n__zeros)), activate(y0))
ISNATLIST(n__cons(n__zeros, y0)) → U51¹(isNat(n__zeros), activate(y0))

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ RuleRemovalProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                        ↳ 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

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                        ↳ QDP

                          ↳ QDP

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

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

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ RuleRemovalProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                        ↳ 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

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                        ↳ QDP

                          ↳ QDP

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

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

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

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

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ RuleRemovalProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                        ↳ 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

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                        ↳ QDP

                          ↳ QDP

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

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

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

ISNATLIST(n__cons(n__0, y0)) → U51¹(isNat(n__0), activate(y0))

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ RuleRemovalProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                        ↳ 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

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                        ↳ QDP

                          ↳ QDP

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

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

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

ISNATLIST(n__cons(n__s(x0), y1)) → U51¹(isNat(n__s(x0)), activate(y1))

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ RuleRemovalProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                        ↳ 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

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ Narrowing

                                        ↳ QDP

                          ↳ QDP

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

ISNATLIST(n__cons(n__length(x0), y1)) → U51¹(isNat(length(x0)), activate(y1))
ISNATLIST(n__cons(x0, y1)) → U51¹(isNat(x0), activate(y1))
U51¹(tt, n__length(x0)) → ISNATLIST(length(x0))
U51¹(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
U51¹(tt, x0) → ISNATLIST(x0)
U51¹(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
ISNATLIST(n__cons(n__0, y0)) → U51¹(isNat(n__0), activate(y0))
ISNATLIST(n__cons(n__zeros, y0)) → U51¹(isNat(cons(0, n__zeros)), activate(y0))
ISNATLIST(n__cons(n__s(x0), y1)) → U51¹(isNat(n__s(x0)), activate(y1))
U51¹(tt, n__zeros) → ISNATLIST(cons(n__0, n__zeros))

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

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

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ RuleRemovalProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                        ↳ 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

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ Narrowing

                                                                                                                            ↳ QDP

                                                                                                                              ↳ DependencyGraphProof

                                        ↳ QDP

                          ↳ QDP

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

ISNATLIST(n__cons(n__length(x0), y1)) → U51¹(isNat(length(x0)), activate(y1))
ISNATLIST(n__cons(x0, y1)) → U51¹(isNat(x0), activate(y1))
ISNATLIST(n__cons(n__zeros, y0)) → U51¹(isNat(n__cons(0, n__zeros)), activate(y0))
U51¹(tt, n__length(x0)) → ISNATLIST(length(x0))
U51¹(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
U51¹(tt, x0) → ISNATLIST(x0)
U51¹(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(n__0, y0)) → U51¹(isNat(n__0), activate(y0))
ISNATLIST(n__cons(n__zeros, y0)) → U51¹(isNat(cons(n__0, n__zeros)), activate(y0))
U51¹(tt, n__zeros) → ISNATLIST(cons(n__0, n__zeros))
ISNATLIST(n__cons(n__s(x0), y1)) → U51¹(isNat(n__s(x0)), activate(y1))

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ RuleRemovalProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                        ↳ 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

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ Narrowing

                                                                                                                            ↳ QDP

                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ Narrowing

                                        ↳ QDP

                          ↳ QDP

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

ISNATLIST(n__cons(n__length(x0), y1)) → U51¹(isNat(length(x0)), activate(y1))
ISNATLIST(n__cons(x0, y1)) → U51¹(isNat(x0), activate(y1))
U51¹(tt, n__length(x0)) → ISNATLIST(length(x0))
U51¹(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
U51¹(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
U51¹(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(n__0, y0)) → U51¹(isNat(n__0), activate(y0))
ISNATLIST(n__cons(n__zeros, y0)) → U51¹(isNat(cons(n__0, n__zeros)), activate(y0))
U51¹(tt, n__zeros) → ISNATLIST(cons(n__0, n__zeros))
ISNATLIST(n__cons(n__s(x0), y1)) → U51¹(isNat(n__s(x0)), activate(y1))

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

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

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ RuleRemovalProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                        ↳ 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

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ Narrowing

                                                                                                                            ↳ QDP

                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ Narrowing

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ Narrowing

                                        ↳ QDP

                          ↳ QDP

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

ISNATLIST(n__cons(n__length(x0), y1)) → U51¹(isNat(length(x0)), activate(y1))
ISNATLIST(n__cons(x0, y1)) → U51¹(isNat(x0), activate(y1))
U51¹(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))
U51¹(tt, n__length(x0)) → ISNATLIST(length(x0))
U51¹(tt, x0) → ISNATLIST(x0)
U51¹(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
U51¹(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(n__0, y0)) → U51¹(isNat(n__0), activate(y0))
ISNATLIST(n__cons(n__zeros, y0)) → U51¹(isNat(cons(n__0, n__zeros)), activate(y0))
ISNATLIST(n__cons(n__s(x0), y1)) → U51¹(isNat(n__s(x0)), activate(y1))

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

ISNATLIST(n__cons(n__zeros, y0)) → U51¹(isNat(n__cons(n__0, n__zeros)), activate(y0))

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ RuleRemovalProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                        ↳ 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

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ Narrowing

                                                                                                                            ↳ QDP

                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ Narrowing

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ Narrowing

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ DependencyGraphProof

                                        ↳ QDP

                          ↳ QDP

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

ISNATLIST(n__cons(n__zeros, y0)) → U51¹(isNat(n__cons(n__0, n__zeros)), activate(y0))
ISNATLIST(n__cons(n__length(x0), y1)) → U51¹(isNat(length(x0)), activate(y1))
ISNATLIST(n__cons(x0, y1)) → U51¹(isNat(x0), activate(y1))
U51¹(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))
U51¹(tt, n__length(x0)) → ISNATLIST(length(x0))
U51¹(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
U51¹(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
U51¹(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(n__0, y0)) → U51¹(isNat(n__0), activate(y0))
ISNATLIST(n__cons(n__s(x0), y1)) → U51¹(isNat(n__s(x0)), activate(y1))

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ RuleRemovalProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                        ↳ 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

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ Narrowing

                                                                                                                            ↳ QDP

                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ Narrowing

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ Narrowing

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ RuleRemovalProof

                                        ↳ QDP

                          ↳ QDP

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

ISNATLIST(n__cons(n__length(x0), y1)) → U51¹(isNat(length(x0)), activate(y1))
ISNATLIST(n__cons(x0, y1)) → U51¹(isNat(x0), activate(y1))
U51¹(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))
U51¹(tt, n__length(x0)) → ISNATLIST(length(x0))
U51¹(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
U51¹(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
U51¹(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(n__0, y0)) → U51¹(isNat(n__0), activate(y0))
ISNATLIST(n__cons(n__s(x0), y1)) → U51¹(isNat(n__s(x0)), activate(y1))

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

ISNATLIST(n__cons(n__length(x0), y1)) → U51¹(isNat(length(x0)), activate(y1))

Used ordering: POLO with Polynomial interpretation [25]:

POL(0) = 0
POL(ISNATLIST(x₁)) = x₁
POL(U21(x₁)) = x₁
POL(U51(x₁, x₂)) = x₁ + 2·x₂
POL(U51¹(x₁, x₂)) = x₁ + x₂
POL(U52(x₁)) = 2·x₁
POL(U61(x₁, x₂, x₃)) = 1 + 2·x₁ + 2·x₂ + 2·x₃
POL(U62(x₁, x₂)) = 1 + x₁ + 2·x₂
POL(activate(x₁)) = x₁
POL(cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(isNat(x₁)) = x₁
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = 1 + 2·x₁
POL(n__0) = 0
POL(n__cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(n__length(x₁)) = 1 + 2·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

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ RuleRemovalProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                        ↳ 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

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ Narrowing

                                                                                                                            ↳ QDP

                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ Narrowing

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ Narrowing

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ RuleRemovalProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ RuleRemovalProof

                                        ↳ QDP

                          ↳ QDP

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

ISNATLIST(n__cons(x0, y1)) → U51¹(isNat(x0), activate(y1))
U51¹(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))
U51¹(tt, n__length(x0)) → ISNATLIST(length(x0))
U51¹(tt, x0) → ISNATLIST(x0)
U51¹(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
U51¹(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(n__0, y0)) → U51¹(isNat(n__0), activate(y0))
ISNATLIST(n__cons(n__s(x0), y1)) → U51¹(isNat(n__s(x0)), activate(y1))

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

U51¹(tt, n__length(x0)) → ISNATLIST(length(x0))

Used ordering: POLO with Polynomial interpretation [25]:

POL(0) = 0
POL(ISNATLIST(x₁)) = x₁
POL(U21(x₁)) = x₁
POL(U51(x₁, x₂)) = x₁ + 2·x₂
POL(U51¹(x₁, x₂)) = 2·x₁ + 2·x₂
POL(U52(x₁)) = x₁
POL(U61(x₁, x₂, x₃)) = 2 + x₁ + 2·x₂ + 2·x₃
POL(U62(x₁, x₂)) = 2 + x₁ + 2·x₂
POL(activate(x₁)) = x₁
POL(cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(isNat(x₁)) = x₁
POL(isNatList(x₁)) = 2·x₁
POL(length(x₁)) = 2 + 2·x₁
POL(n__0) = 0
POL(n__cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(n__length(x₁)) = 2 + 2·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

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ RuleRemovalProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                        ↳ 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

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ Narrowing

                                                                                                                            ↳ QDP

                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ Narrowing

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ Narrowing

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ RuleRemovalProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ RuleRemovalProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ QDPOrderProof

                                        ↳ QDP

                          ↳ QDP

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

ISNATLIST(n__cons(x0, y1)) → U51¹(isNat(x0), activate(y1))
U51¹(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))
U51¹(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
U51¹(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
U51¹(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(n__0, y0)) → U51¹(isNat(n__0), activate(y0))
ISNATLIST(n__cons(n__s(x0), y1)) → U51¹(isNat(n__s(x0)), activate(y1))

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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.

ISNATLIST(n__cons(n__s(x0), y1)) → U51¹(isNat(n__s(x0)), activate(y1))

The remaining pairs can at least be oriented weakly.

ISNATLIST(n__cons(x0, y1)) → U51¹(isNat(x0), activate(y1))
U51¹(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))
U51¹(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
U51¹(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
U51¹(tt, x0) → ISNATLIST(x0)
ISNATLIST(n__cons(n__0, y0)) → U51¹(isNat(n__0), activate(y0))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(ISNATLIST(x₁)) = 1 + x₁
POL(U21(x₁)) = 1
POL(U51(x₁, x₂)) = x₂
POL(U51¹(x₁, x₂)) = 1 + x₂
POL(U52(x₁)) = 0
POL(U61(x₁, x₂, x₃)) = 1 + x₂
POL(U62(x₁, x₂)) = 1 + x₂
POL(activate(x₁)) = x₁
POL(cons(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = 1 + x₁
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = 1 + x₁
POL(n__0) = 0
POL(n__cons(x₁, x₂)) = x₁ + x₂
POL(n__length(x₁)) = 1 + x₁
POL(n__nil) = 0
POL(n__s(x₁)) = 1
POL(n__zeros) = 1
POL(nil) = 0
POL(s(x₁)) = 1
POL(tt) = 0
POL(zeros) = 1

The following usable rules [17] were oriented:

U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
length(X) → n__length(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
zeros → n__zeros
zeros → cons(0, n__zeros)
0 → n__0
U52(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U21(tt) → tt
U62(tt, L) → s(length(activate(L)))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
activate(n__nil) → nil
activate(X) → X
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__zeros) → zeros

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ RuleRemovalProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                        ↳ 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

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ Narrowing

                                                                                                                            ↳ QDP

                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ Narrowing

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ Narrowing

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ RuleRemovalProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ RuleRemovalProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ QDPOrderProof

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ NonTerminationProof

                                        ↳ QDP

                          ↳ QDP

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

ISNATLIST(n__cons(x0, y1)) → U51¹(isNat(x0), activate(y1))
U51¹(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))
U51¹(tt, x0) → ISNATLIST(x0)
U51¹(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
U51¹(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(n__0, y0)) → U51¹(isNat(n__0), activate(y0))

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

The TRS P consists of the following rules:

ISNATLIST(n__cons(x0, y1)) → U51¹(isNat(x0), activate(y1))
U51¹(tt, n__zeros) → ISNATLIST(n__cons(n__0, n__zeros))
U51¹(tt, x0) → ISNATLIST(x0)
U51¹(tt, n__cons(x0, x1)) → ISNATLIST(n__cons(x0, x1))
U51¹(tt, n__zeros) → ISNATLIST(n__cons(0, n__zeros))
ISNATLIST(n__cons(n__0, y0)) → U51¹(isNat(n__0), activate(y0))

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

s = U51¹(isNat(n__0), activate(n__zeros)) evaluates to t =U51¹(isNat(n__0), activate(n__zeros))

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

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

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

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

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

ISNATLIST(n__cons(n__0, n__zeros)) → U51¹(isNat(n__0), activate(n__zeros))
with rule ISNATLIST(n__cons(x0, y1)) → U51¹(isNat(x0), activate(y1))

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

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

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ RuleRemovalProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                        ↳ QDP

                                        ↳ QDP

                          ↳ QDP

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

U61¹(tt, L, N) → U62¹(isNat(activate(N)), activate(L))
LENGTH(cons(N, L)) → U61¹(isNatList(activate(L)), activate(L), N)
U62¹(tt, L) → LENGTH(activate(L))

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U62(tt, L) → s(length(activate(L)))
U21(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

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

U41¹(tt, V2) → ISNATILIST(activate(V2))
ISNATILIST(n__cons(V1, V2)) → U41¹(isNat(activate(V1)), activate(V2))

The TRS R consists of the following rules:

zeros → cons(0, n__zeros)
U21(tt) → tt
U41(tt, V2) → U42(isNatIList(activate(V2)))
U42(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNatIList(n__cons(V1, V2)) → U41(isNat(activate(V1)), activate(V2))
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X

POL(0) = 0
POL(ISNATILIST(x₁)) = x₁
POL(U21(x₁)) = x₁
POL(U41¹(x₁, x₂)) = x₁ + x₂
POL(U51(x₁, x₂)) = x₁ + 2·x₂
POL(U52(x₁)) = x₁
POL(U61(x₁, x₂, x₃)) = x₁ + 2·x₂ + 2·x₃
POL(U62(x₁, x₂)) = x₁ + 2·x₂
POL(activate(x₁)) = x₁
POL(cons(x₁, x₂)) = x₁ + 2·x₂
POL(isNat(x₁)) = x₁
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = 2·x₁
POL(n__0) = 0
POL(n__cons(x₁, x₂)) = x₁ + 2·x₂
POL(n__length(x₁)) = 2·x₁
POL(n__nil) = 2
POL(n__s(x₁)) = x₁
POL(n__zeros) = 0
POL(nil) = 2
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ Narrowing

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

U41¹(tt, V2) → ISNATILIST(activate(V2))
ISNATILIST(n__cons(V1, V2)) → U41¹(isNat(activate(V1)), activate(V2))

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U41¹(tt, V2) → ISNATILIST(activate(V2)) at position [0] we obtained the following new rules:

U41¹(tt, n__zeros) → ISNATILIST(zeros)
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(x0, x1))
U41¹(tt, n__0) → ISNATILIST(0)
U41¹(tt, n__nil) → ISNATILIST(nil)
U41¹(tt, n__s(x0)) → ISNATILIST(s(x0))
U41¹(tt, x0) → ISNATILIST(x0)
U41¹(tt, n__length(x0)) → ISNATILIST(length(x0))

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ Narrowing

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

U41¹(tt, n__zeros) → ISNATILIST(zeros)
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(x0, x1))
U41¹(tt, n__nil) → ISNATILIST(nil)
U41¹(tt, n__0) → ISNATILIST(0)
U41¹(tt, x0) → ISNATILIST(x0)
U41¹(tt, n__s(x0)) → ISNATILIST(s(x0))
ISNATILIST(n__cons(V1, V2)) → U41¹(isNat(activate(V1)), activate(V2))
U41¹(tt, n__length(x0)) → ISNATILIST(length(x0))

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ISNATILIST(n__cons(V1, V2)) → U41¹(isNat(activate(V1)), activate(V2)) at position [0] we obtained the following new rules:

ISNATILIST(n__cons(n__0, y1)) → U41¹(isNat(0), activate(y1))
ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(s(x0)), activate(y1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U41¹(isNat(cons(x0, x1)), activate(y1))
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
ISNATILIST(n__cons(n__length(x0), y1)) → U41¹(isNat(length(x0)), activate(y1))
ISNATILIST(n__cons(n__nil, y1)) → U41¹(isNat(nil), activate(y1))
ISNATILIST(n__cons(n__zeros, y1)) → U41¹(isNat(zeros), activate(y1))

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

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

ISNATILIST(n__cons(n__0, y1)) → U41¹(isNat(0), activate(y1))
ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(s(x0)), activate(y1))
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
U41¹(tt, n__0) → ISNATILIST(0)
U41¹(tt, n__length(x0)) → ISNATILIST(length(x0))
ISNATILIST(n__cons(n__zeros, y1)) → U41¹(isNat(zeros), activate(y1))
U41¹(tt, n__zeros) → ISNATILIST(zeros)
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U41¹(isNat(cons(x0, x1)), activate(y1))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(x0, x1))
U41¹(tt, n__nil) → ISNATILIST(nil)
ISNATILIST(n__cons(n__length(x0), y1)) → U41¹(isNat(length(x0)), activate(y1))
U41¹(tt, x0) → ISNATILIST(x0)
U41¹(tt, n__s(x0)) → ISNATILIST(s(x0))
ISNATILIST(n__cons(n__nil, y1)) → U41¹(isNat(nil), activate(y1))

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U41¹(tt, n__zeros) → ISNATILIST(zeros) at position [0] we obtained the following new rules:

U41¹(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
U41¹(tt, n__zeros) → ISNATILIST(n__zeros)

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

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

U41¹(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
ISNATILIST(n__cons(n__0, y1)) → U41¹(isNat(0), activate(y1))
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(s(x0)), activate(y1))
U41¹(tt, n__0) → ISNATILIST(0)
U41¹(tt, n__length(x0)) → ISNATILIST(length(x0))
ISNATILIST(n__cons(n__zeros, y1)) → U41¹(isNat(zeros), activate(y1))
U41¹(tt, n__zeros) → ISNATILIST(n__zeros)
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U41¹(isNat(cons(x0, x1)), activate(y1))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(x0, x1))
ISNATILIST(n__cons(n__length(x0), y1)) → U41¹(isNat(length(x0)), activate(y1))
U41¹(tt, n__nil) → ISNATILIST(nil)
U41¹(tt, n__s(x0)) → ISNATILIST(s(x0))
U41¹(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(n__nil, y1)) → U41¹(isNat(nil), activate(y1))

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

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

U41¹(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
ISNATILIST(n__cons(n__0, y1)) → U41¹(isNat(0), activate(y1))
ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(s(x0)), activate(y1))
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
U41¹(tt, n__0) → ISNATILIST(0)
U41¹(tt, n__length(x0)) → ISNATILIST(length(x0))
ISNATILIST(n__cons(n__zeros, y1)) → U41¹(isNat(zeros), activate(y1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U41¹(isNat(cons(x0, x1)), activate(y1))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(x0, x1))
U41¹(tt, n__nil) → ISNATILIST(nil)
ISNATILIST(n__cons(n__length(x0), y1)) → U41¹(isNat(length(x0)), activate(y1))
U41¹(tt, n__s(x0)) → ISNATILIST(s(x0))
U41¹(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(n__nil, y1)) → U41¹(isNat(nil), activate(y1))

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ISNATILIST(n__cons(n__0, y1)) → U41¹(isNat(0), activate(y1)) at position [0] we obtained the following new rules:

ISNATILIST(n__cons(n__0, y0)) → U41¹(isNat(n__0), activate(y0))

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Narrowing

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

U41¹(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(s(x0)), activate(y1))
U41¹(tt, n__0) → ISNATILIST(0)
U41¹(tt, n__length(x0)) → ISNATILIST(length(x0))
ISNATILIST(n__cons(n__zeros, y1)) → U41¹(isNat(zeros), activate(y1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U41¹(isNat(cons(x0, x1)), activate(y1))
ISNATILIST(n__cons(n__0, y0)) → U41¹(isNat(n__0), activate(y0))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(x0, x1))
ISNATILIST(n__cons(n__length(x0), y1)) → U41¹(isNat(length(x0)), activate(y1))
U41¹(tt, n__nil) → ISNATILIST(nil)
U41¹(tt, x0) → ISNATILIST(x0)
U41¹(tt, n__s(x0)) → ISNATILIST(s(x0))
ISNATILIST(n__cons(n__nil, y1)) → U41¹(isNat(nil), activate(y1))

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U41¹(tt, n__cons(x0, x1)) → ISNATILIST(cons(x0, x1)) at position [0] we obtained the following new rules:

U41¹(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(x0, x1))

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ Narrowing

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

U41¹(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(s(x0)), activate(y1))
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
U41¹(tt, n__0) → ISNATILIST(0)
U41¹(tt, n__length(x0)) → ISNATILIST(length(x0))
ISNATILIST(n__cons(n__zeros, y1)) → U41¹(isNat(zeros), activate(y1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U41¹(isNat(cons(x0, x1)), activate(y1))
ISNATILIST(n__cons(n__0, y0)) → U41¹(isNat(n__0), activate(y0))
U41¹(tt, n__nil) → ISNATILIST(nil)
ISNATILIST(n__cons(n__length(x0), y1)) → U41¹(isNat(length(x0)), activate(y1))
U41¹(tt, n__s(x0)) → ISNATILIST(s(x0))
U41¹(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(n__nil, y1)) → U41¹(isNat(nil), activate(y1))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(x0, x1))

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(s(x0)), activate(y1)) at position [0] we obtained the following new rules:

ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(n__s(x0)), activate(y1))

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Narrowing

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

U41¹(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(n__s(x0)), activate(y1))
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
U41¹(tt, n__0) → ISNATILIST(0)
U41¹(tt, n__length(x0)) → ISNATILIST(length(x0))
ISNATILIST(n__cons(n__zeros, y1)) → U41¹(isNat(zeros), activate(y1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U41¹(isNat(cons(x0, x1)), activate(y1))
ISNATILIST(n__cons(n__0, y0)) → U41¹(isNat(n__0), activate(y0))
ISNATILIST(n__cons(n__length(x0), y1)) → U41¹(isNat(length(x0)), activate(y1))
U41¹(tt, n__nil) → ISNATILIST(nil)
U41¹(tt, x0) → ISNATILIST(x0)
U41¹(tt, n__s(x0)) → ISNATILIST(s(x0))
ISNATILIST(n__cons(n__nil, y1)) → U41¹(isNat(nil), activate(y1))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(x0, x1))

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U41¹(tt, n__0) → ISNATILIST(0) at position [0] we obtained the following new rules:

U41¹(tt, n__0) → ISNATILIST(n__0)

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

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

U41¹(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(n__s(x0)), activate(y1))
U41¹(tt, n__0) → ISNATILIST(n__0)
U41¹(tt, n__length(x0)) → ISNATILIST(length(x0))
ISNATILIST(n__cons(n__zeros, y1)) → U41¹(isNat(zeros), activate(y1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U41¹(isNat(cons(x0, x1)), activate(y1))
ISNATILIST(n__cons(n__0, y0)) → U41¹(isNat(n__0), activate(y0))
U41¹(tt, n__nil) → ISNATILIST(nil)
ISNATILIST(n__cons(n__length(x0), y1)) → U41¹(isNat(length(x0)), activate(y1))
U41¹(tt, n__s(x0)) → ISNATILIST(s(x0))
U41¹(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(n__nil, y1)) → U41¹(isNat(nil), activate(y1))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(x0, x1))

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

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

U41¹(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(n__s(x0)), activate(y1))
U41¹(tt, n__length(x0)) → ISNATILIST(length(x0))
ISNATILIST(n__cons(n__zeros, y1)) → U41¹(isNat(zeros), activate(y1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U41¹(isNat(cons(x0, x1)), activate(y1))
ISNATILIST(n__cons(n__0, y0)) → U41¹(isNat(n__0), activate(y0))
U41¹(tt, n__nil) → ISNATILIST(nil)
ISNATILIST(n__cons(n__length(x0), y1)) → U41¹(isNat(length(x0)), activate(y1))
U41¹(tt, n__s(x0)) → ISNATILIST(s(x0))
U41¹(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(n__nil, y1)) → U41¹(isNat(nil), activate(y1))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(x0, x1))

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U41¹(tt, n__nil) → ISNATILIST(nil) at position [0] we obtained the following new rules:

U41¹(tt, n__nil) → ISNATILIST(n__nil)

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

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

U41¹(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(n__s(x0)), activate(y1))
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
U41¹(tt, n__nil) → ISNATILIST(n__nil)
U41¹(tt, n__length(x0)) → ISNATILIST(length(x0))
ISNATILIST(n__cons(n__zeros, y1)) → U41¹(isNat(zeros), activate(y1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U41¹(isNat(cons(x0, x1)), activate(y1))
ISNATILIST(n__cons(n__0, y0)) → U41¹(isNat(n__0), activate(y0))
ISNATILIST(n__cons(n__length(x0), y1)) → U41¹(isNat(length(x0)), activate(y1))
U41¹(tt, x0) → ISNATILIST(x0)
U41¹(tt, n__s(x0)) → ISNATILIST(s(x0))
ISNATILIST(n__cons(n__nil, y1)) → U41¹(isNat(nil), activate(y1))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(x0, x1))

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

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

U41¹(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(n__s(x0)), activate(y1))
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U41¹(isNat(cons(x0, x1)), activate(y1))
ISNATILIST(n__cons(n__0, y0)) → U41¹(isNat(n__0), activate(y0))
ISNATILIST(n__cons(n__length(x0), y1)) → U41¹(isNat(length(x0)), activate(y1))
U41¹(tt, x0) → ISNATILIST(x0)
U41¹(tt, n__s(x0)) → ISNATILIST(s(x0))
ISNATILIST(n__cons(n__nil, y1)) → U41¹(isNat(nil), activate(y1))
U41¹(tt, n__length(x0)) → ISNATILIST(length(x0))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(x0, x1))
ISNATILIST(n__cons(n__zeros, y1)) → U41¹(isNat(zeros), activate(y1))

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U41¹(tt, n__s(x0)) → ISNATILIST(s(x0)) at position [0] we obtained the following new rules:

U41¹(tt, n__s(x0)) → ISNATILIST(n__s(x0))

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

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

U41¹(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(n__s(x0)), activate(y1))
U41¹(tt, n__length(x0)) → ISNATILIST(length(x0))
ISNATILIST(n__cons(n__zeros, y1)) → U41¹(isNat(zeros), activate(y1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U41¹(isNat(cons(x0, x1)), activate(y1))
ISNATILIST(n__cons(n__0, y0)) → U41¹(isNat(n__0), activate(y0))
ISNATILIST(n__cons(n__length(x0), y1)) → U41¹(isNat(length(x0)), activate(y1))
U41¹(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(n__nil, y1)) → U41¹(isNat(nil), activate(y1))
U41¹(tt, n__s(x0)) → ISNATILIST(n__s(x0))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(x0, x1))

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

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

U41¹(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(n__s(x0)), activate(y1))
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U41¹(isNat(cons(x0, x1)), activate(y1))
ISNATILIST(n__cons(n__0, y0)) → U41¹(isNat(n__0), activate(y0))
ISNATILIST(n__cons(n__length(x0), y1)) → U41¹(isNat(length(x0)), activate(y1))
U41¹(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(n__nil, y1)) → U41¹(isNat(nil), activate(y1))
U41¹(tt, n__length(x0)) → ISNATILIST(length(x0))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(x0, x1))
ISNATILIST(n__cons(n__zeros, y1)) → U41¹(isNat(zeros), activate(y1))

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ISNATILIST(n__cons(n__cons(x0, x1), y1)) → U41¹(isNat(cons(x0, x1)), activate(y1)) at position [0] we obtained the following new rules:

ISNATILIST(n__cons(n__cons(x0, x1), y2)) → U41¹(isNat(n__cons(x0, x1)), activate(y2))

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

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

U41¹(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(n__s(x0)), activate(y1))
ISNATILIST(n__cons(n__0, y0)) → U41¹(isNat(n__0), activate(y0))
ISNATILIST(n__cons(n__length(x0), y1)) → U41¹(isNat(length(x0)), activate(y1))
U41¹(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(n__nil, y1)) → U41¹(isNat(nil), activate(y1))
U41¹(tt, n__length(x0)) → ISNATILIST(length(x0))
ISNATILIST(n__cons(n__cons(x0, x1), y2)) → U41¹(isNat(n__cons(x0, x1)), activate(y2))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(x0, x1))
ISNATILIST(n__cons(n__zeros, y1)) → U41¹(isNat(zeros), activate(y1))

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Narrowing

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

U41¹(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(n__s(x0)), activate(y1))
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
ISNATILIST(n__cons(n__0, y0)) → U41¹(isNat(n__0), activate(y0))
ISNATILIST(n__cons(n__length(x0), y1)) → U41¹(isNat(length(x0)), activate(y1))
U41¹(tt, x0) → ISNATILIST(x0)
ISNATILIST(n__cons(n__nil, y1)) → U41¹(isNat(nil), activate(y1))
U41¹(tt, n__length(x0)) → ISNATILIST(length(x0))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(x0, x1))
ISNATILIST(n__cons(n__zeros, y1)) → U41¹(isNat(zeros), activate(y1))

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ISNATILIST(n__cons(n__nil, y1)) → U41¹(isNat(nil), activate(y1)) at position [0] we obtained the following new rules:

ISNATILIST(n__cons(n__nil, y0)) → U41¹(isNat(n__nil), activate(y0))

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Narrowing

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

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

U41¹(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(n__s(x0)), activate(y1))
ISNATILIST(n__cons(n__0, y0)) → U41¹(isNat(n__0), activate(y0))
ISNATILIST(n__cons(n__length(x0), y1)) → U41¹(isNat(length(x0)), activate(y1))
U41¹(tt, x0) → ISNATILIST(x0)
U41¹(tt, n__length(x0)) → ISNATILIST(length(x0))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(x0, x1))
ISNATILIST(n__cons(n__nil, y0)) → U41¹(isNat(n__nil), activate(y0))
ISNATILIST(n__cons(n__zeros, y1)) → U41¹(isNat(zeros), activate(y1))

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Narrowing

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

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

U41¹(tt, n__zeros) → ISNATILIST(cons(0, n__zeros))
ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(n__s(x0)), activate(y1))
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
ISNATILIST(n__cons(n__0, y0)) → U41¹(isNat(n__0), activate(y0))
ISNATILIST(n__cons(n__length(x0), y1)) → U41¹(isNat(length(x0)), activate(y1))
U41¹(tt, x0) → ISNATILIST(x0)
U41¹(tt, n__length(x0)) → ISNATILIST(length(x0))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(x0, x1))
ISNATILIST(n__cons(n__zeros, y1)) → U41¹(isNat(zeros), activate(y1))

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U41¹(tt, n__zeros) → ISNATILIST(cons(0, n__zeros)) at position [0] we obtained the following new rules:

U41¹(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U41¹(tt, n__zeros) → ISNATILIST(cons(n__0, n__zeros))

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Narrowing

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ Narrowing

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

ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(n__s(x0)), activate(y1))
ISNATILIST(n__cons(n__0, y0)) → U41¹(isNat(n__0), activate(y0))
U41¹(tt, n__zeros) → ISNATILIST(cons(n__0, n__zeros))
ISNATILIST(n__cons(n__length(x0), y1)) → U41¹(isNat(length(x0)), activate(y1))
U41¹(tt, x0) → ISNATILIST(x0)
U41¹(tt, n__length(x0)) → ISNATILIST(length(x0))
U41¹(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(x0, x1))
ISNATILIST(n__cons(n__zeros, y1)) → U41¹(isNat(zeros), activate(y1))

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ISNATILIST(n__cons(n__zeros, y1)) → U41¹(isNat(zeros), activate(y1)) at position [0] we obtained the following new rules:

ISNATILIST(n__cons(n__zeros, y0)) → U41¹(isNat(n__zeros), activate(y0))
ISNATILIST(n__cons(n__zeros, y0)) → U41¹(isNat(cons(0, n__zeros)), activate(y0))

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Narrowing

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ Narrowing

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

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

ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(n__s(x0)), activate(y1))
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
ISNATILIST(n__cons(n__zeros, y0)) → U41¹(isNat(n__zeros), activate(y0))
ISNATILIST(n__cons(n__0, y0)) → U41¹(isNat(n__0), activate(y0))
ISNATILIST(n__cons(n__zeros, y0)) → U41¹(isNat(cons(0, n__zeros)), activate(y0))
ISNATILIST(n__cons(n__length(x0), y1)) → U41¹(isNat(length(x0)), activate(y1))
U41¹(tt, n__zeros) → ISNATILIST(cons(n__0, n__zeros))
U41¹(tt, x0) → ISNATILIST(x0)
U41¹(tt, n__length(x0)) → ISNATILIST(length(x0))
U41¹(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(x0, x1))

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Narrowing

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ Narrowing

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Narrowing

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

ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(n__s(x0)), activate(y1))
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
ISNATILIST(n__cons(n__0, y0)) → U41¹(isNat(n__0), activate(y0))
ISNATILIST(n__cons(n__zeros, y0)) → U41¹(isNat(cons(0, n__zeros)), activate(y0))
U41¹(tt, n__zeros) → ISNATILIST(cons(n__0, n__zeros))
ISNATILIST(n__cons(n__length(x0), y1)) → U41¹(isNat(length(x0)), activate(y1))
U41¹(tt, x0) → ISNATILIST(x0)
U41¹(tt, n__length(x0)) → ISNATILIST(length(x0))
U41¹(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(x0, x1))

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ISNATILIST(n__cons(n__zeros, y0)) → U41¹(isNat(cons(0, n__zeros)), activate(y0)) at position [0] we obtained the following new rules:

ISNATILIST(n__cons(n__zeros, y0)) → U41¹(isNat(n__cons(0, n__zeros)), activate(y0))
ISNATILIST(n__cons(n__zeros, y0)) → U41¹(isNat(cons(n__0, n__zeros)), activate(y0))

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Narrowing

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ Narrowing

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Narrowing

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

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

ISNATILIST(n__cons(n__zeros, y0)) → U41¹(isNat(n__cons(0, n__zeros)), activate(y0))
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(n__s(x0)), activate(y1))
ISNATILIST(n__cons(n__0, y0)) → U41¹(isNat(n__0), activate(y0))
ISNATILIST(n__cons(n__length(x0), y1)) → U41¹(isNat(length(x0)), activate(y1))
U41¹(tt, n__zeros) → ISNATILIST(cons(n__0, n__zeros))
U41¹(tt, x0) → ISNATILIST(x0)
U41¹(tt, n__length(x0)) → ISNATILIST(length(x0))
U41¹(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(x0, x1))
ISNATILIST(n__cons(n__zeros, y0)) → U41¹(isNat(cons(n__0, n__zeros)), activate(y0))

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Narrowing

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ Narrowing

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Narrowing

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

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

ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(n__s(x0)), activate(y1))
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
ISNATILIST(n__cons(n__0, y0)) → U41¹(isNat(n__0), activate(y0))
U41¹(tt, n__zeros) → ISNATILIST(cons(n__0, n__zeros))
ISNATILIST(n__cons(n__length(x0), y1)) → U41¹(isNat(length(x0)), activate(y1))
U41¹(tt, x0) → ISNATILIST(x0)
U41¹(tt, n__length(x0)) → ISNATILIST(length(x0))
U41¹(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(x0, x1))
ISNATILIST(n__cons(n__zeros, y0)) → U41¹(isNat(cons(n__0, n__zeros)), activate(y0))

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U41¹(tt, n__zeros) → ISNATILIST(cons(n__0, n__zeros)) at position [0] we obtained the following new rules:

U41¹(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Narrowing

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ Narrowing

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Narrowing

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Narrowing

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

ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(n__s(x0)), activate(y1))
ISNATILIST(n__cons(n__0, y0)) → U41¹(isNat(n__0), activate(y0))
U41¹(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))
ISNATILIST(n__cons(n__length(x0), y1)) → U41¹(isNat(length(x0)), activate(y1))
U41¹(tt, x0) → ISNATILIST(x0)
U41¹(tt, n__length(x0)) → ISNATILIST(length(x0))
U41¹(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(x0, x1))
ISNATILIST(n__cons(n__zeros, y0)) → U41¹(isNat(cons(n__0, n__zeros)), activate(y0))

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ISNATILIST(n__cons(n__zeros, y0)) → U41¹(isNat(cons(n__0, n__zeros)), activate(y0)) at position [0] we obtained the following new rules:

ISNATILIST(n__cons(n__zeros, y0)) → U41¹(isNat(n__cons(n__0, n__zeros)), activate(y0))

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Narrowing

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ Narrowing

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Narrowing

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Narrowing

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

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

ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(n__s(x0)), activate(y1))
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
ISNATILIST(n__cons(n__0, y0)) → U41¹(isNat(n__0), activate(y0))
ISNATILIST(n__cons(n__zeros, y0)) → U41¹(isNat(n__cons(n__0, n__zeros)), activate(y0))
ISNATILIST(n__cons(n__length(x0), y1)) → U41¹(isNat(length(x0)), activate(y1))
U41¹(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))
U41¹(tt, x0) → ISNATILIST(x0)
U41¹(tt, n__length(x0)) → ISNATILIST(length(x0))
U41¹(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(x0, x1))

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Narrowing

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ Narrowing

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Narrowing

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Narrowing

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ RuleRemovalProof

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

ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(n__s(x0)), activate(y1))
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
ISNATILIST(n__cons(n__0, y0)) → U41¹(isNat(n__0), activate(y0))
U41¹(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))
ISNATILIST(n__cons(n__length(x0), y1)) → U41¹(isNat(length(x0)), activate(y1))
U41¹(tt, x0) → ISNATILIST(x0)
U41¹(tt, n__length(x0)) → ISNATILIST(length(x0))
U41¹(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(x0, x1))

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

U41¹(tt, n__length(x0)) → ISNATILIST(length(x0))

Used ordering: POLO with Polynomial interpretation [25]:

POL(0) = 0
POL(ISNATILIST(x₁)) = x₁
POL(U21(x₁)) = x₁
POL(U41¹(x₁, x₂)) = 2·x₁ + 2·x₂
POL(U51(x₁, x₂)) = x₁ + 2·x₂
POL(U52(x₁)) = 2·x₁
POL(U61(x₁, x₂, x₃)) = 1 + 2·x₁ + 2·x₂ + x₃
POL(U62(x₁, x₂)) = 1 + x₁ + 2·x₂
POL(activate(x₁)) = x₁
POL(cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(isNat(x₁)) = x₁
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = 1 + 2·x₁
POL(n__0) = 0
POL(n__cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(n__length(x₁)) = 1 + 2·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

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Narrowing

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ Narrowing

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Narrowing

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Narrowing

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ RuleRemovalProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ RuleRemovalProof

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

ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(n__s(x0)), activate(y1))
ISNATILIST(n__cons(n__0, y0)) → U41¹(isNat(n__0), activate(y0))
ISNATILIST(n__cons(n__length(x0), y1)) → U41¹(isNat(length(x0)), activate(y1))
U41¹(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))
U41¹(tt, x0) → ISNATILIST(x0)
U41¹(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(x0, x1))

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

ISNATILIST(n__cons(n__length(x0), y1)) → U41¹(isNat(length(x0)), activate(y1))

Used ordering: POLO with Polynomial interpretation [25]:

POL(0) = 0
POL(ISNATILIST(x₁)) = 2·x₁
POL(U21(x₁)) = x₁
POL(U41¹(x₁, x₂)) = x₁ + 2·x₂
POL(U51(x₁, x₂)) = x₁ + 2·x₂
POL(U52(x₁)) = 2·x₁
POL(U61(x₁, x₂, x₃)) = 1 + x₁ + x₂ + 2·x₃
POL(U62(x₁, x₂)) = 1 + x₁ + x₂
POL(activate(x₁)) = x₁
POL(cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(isNat(x₁)) = x₁
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = 1 + x₁
POL(n__0) = 0
POL(n__cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(n__length(x₁)) = 1 + x₁
POL(n__nil) = 0
POL(n__s(x₁)) = x₁
POL(n__zeros) = 0
POL(nil) = 0
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Narrowing

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ Narrowing

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Narrowing

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Narrowing

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ RuleRemovalProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ RuleRemovalProof

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ QDPOrderProof

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

ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(n__s(x0)), activate(y1))
ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
ISNATILIST(n__cons(n__0, y0)) → U41¹(isNat(n__0), activate(y0))
U41¹(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))
U41¹(tt, x0) → ISNATILIST(x0)
U41¹(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(x0, x1))

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

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

The following pairs can be oriented strictly and are deleted.

ISNATILIST(n__cons(n__s(x0), y1)) → U41¹(isNat(n__s(x0)), activate(y1))

The remaining pairs can at least be oriented weakly.

ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
ISNATILIST(n__cons(n__0, y0)) → U41¹(isNat(n__0), activate(y0))
U41¹(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))
U41¹(tt, x0) → ISNATILIST(x0)
U41¹(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(x0, x1))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(ISNATILIST(x₁)) = x₁
POL(U21(x₁)) = 0
POL(U41¹(x₁, x₂)) = x₂
POL(U51(x₁, x₂)) = 0
POL(U52(x₁)) = 0
POL(U61(x₁, x₂, x₃)) = 1
POL(U62(x₁, x₂)) = 1
POL(activate(x₁)) = x₁
POL(cons(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = 0
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = 1 + x₁
POL(n__0) = 0
POL(n__cons(x₁, x₂)) = x₁ + x₂
POL(n__length(x₁)) = 1 + x₁
POL(n__nil) = 0
POL(n__s(x₁)) = 1
POL(n__zeros) = 0
POL(nil) = 0
POL(s(x₁)) = 1
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

activate(X) → X
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(n__zeros) → zeros
activate(n__0) → 0
zeros → cons(0, n__zeros)
0 → n__0
U52(tt) → tt
U51(tt, V2) → U52(isNatList(activate(V2)))
zeros → n__zeros
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
nil → n__nil
U62(tt, L) → s(length(activate(L)))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
length(X) → n__length(X)

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ DependencyPairsProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ AND

                          ↳ QDP

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ DependencyGraphProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ DependencyGraphProof

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Narrowing

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ Narrowing

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Narrowing

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Narrowing

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ RuleRemovalProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ RuleRemovalProof

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ NonTerminationProof

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

ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
ISNATILIST(n__cons(n__0, y0)) → U41¹(isNat(n__0), activate(y0))
U41¹(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))
U41¹(tt, x0) → ISNATILIST(x0)
U41¹(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(x0, x1))

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))
ISNATILIST(n__cons(n__0, y0)) → U41¹(isNat(n__0), activate(y0))
U41¹(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))
U41¹(tt, x0) → ISNATILIST(x0)
U41¹(tt, n__zeros) → ISNATILIST(n__cons(0, n__zeros))
U41¹(tt, n__cons(x0, x1)) → ISNATILIST(n__cons(x0, x1))

The TRS R consists of the following rules:

activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__nil) → nil
activate(X) → X
isNat(n__0) → tt
isNat(n__s(V1)) → U21(isNat(activate(V1)))
U21(tt) → tt
nil → n__nil
cons(X1, X2) → n__cons(X1, X2)
s(X) → n__s(X)
length(cons(N, L)) → U61(isNatList(activate(L)), activate(L), N)
length(X) → n__length(X)
isNatList(n__cons(V1, V2)) → U51(isNat(activate(V1)), activate(V2))
U61(tt, L, N) → U62(isNat(activate(N)), activate(L))
U62(tt, L) → s(length(activate(L)))
U51(tt, V2) → U52(isNatList(activate(V2)))
U52(tt) → tt
0 → n__0
zeros → cons(0, n__zeros)
zeros → n__zeros

s = U41¹(isNat(n__0), activate(n__zeros)) evaluates to t =U41¹(isNat(n__0), activate(n__zeros))

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

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

U41¹(isNat(n__0), activate(n__zeros)) → U41¹(isNat(n__0), n__zeros)
with rule activate(X) → X at position [1] and matcher [X / n__zeros]

U41¹(isNat(n__0), n__zeros) → U41¹(tt, n__zeros)
with rule isNat(n__0) → tt at position [0] and matcher [ ]

U41¹(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros))
with rule U41¹(tt, n__zeros) → ISNATILIST(n__cons(n__0, n__zeros)) at position [] and matcher [ ]

ISNATILIST(n__cons(n__0, n__zeros)) → U41¹(isNat(n__0), activate(n__zeros))
with rule ISNATILIST(n__cons(x0, y1)) → U41¹(isNat(x0), activate(y1))

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

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