Termination w.r.t. Q proof of /tmp/tmpzqc21u/OvConsOS_nokinds_noand

Termination w.r.t. Q of the given QTRS could not be shown:

0 QTRS

↳1 DependencyPairsProof (⇔)

↳2 QDP

↳3 DependencyGraphProof (⇔)

↳4 AND

  ↳5 QDP

    ↳6 QDPOrderProof (⇔)

    ↳7 QDP

    ↳8 DependencyGraphProof (⇔)

    ↳9 TRUE

  ↳10 QDP

(0) Obligation:

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

a__zeros → cons(0, zeros)
a__U11(tt) → tt
a__U21(tt) → tt
a__U31(tt) → tt
a__U41(tt, V2) → a__U42(a__isNatIList(V2))
a__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatList(V2))
a__U52(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIList(V2))
a__U62(tt) → tt
a__U71(tt, L, N) → a__U72(a__isNat(N), L)
a__U72(tt, L) → s(a__length(mark(L)))
a__U81(tt) → nil
a__U91(tt, IL, M, N) → a__U92(a__isNat(M), IL, M, N)
a__U92(tt, IL, M, N) → a__U93(a__isNat(N), IL, M, N)
a__U93(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatList(V1))
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNatIList(V) → a__U31(a__isNatList(V))
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNat(V1), V2)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
a__isNatList(take(V1, V2)) → a__U61(a__isNat(V1), V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U71(a__isNatList(L), L, N)
a__take(0, IL) → a__U81(a__isNatIList(IL))
a__take(s(M), cons(N, IL)) → a__U91(a__isNatIList(IL), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X)) → a__U11(mark(X))
mark(U21(X)) → a__U21(mark(X))
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X)) → a__U62(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2)) → a__U72(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(U81(X)) → a__U81(mark(X))
mark(U91(X1, X2, X3, X4)) → a__U91(mark(X1), X2, X3, X4)
mark(U92(X1, X2, X3, X4)) → a__U92(mark(X1), X2, X3, X4)
mark(U93(X1, X2, X3, X4)) → a__U93(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X) → U11(X)
a__U21(X) → U21(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__isNatList(X) → isNatList(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2) → U72(X1, X2)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3, X4) → U91(X1, X2, X3, X4)
a__U92(X1, X2, X3, X4) → U92(X1, X2, X3, X4)
a__U93(X1, X2, X3, X4) → U93(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(2) Obligation:

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

A__U41(tt, V2) → A__U42(a__isNatIList(V2))
A__U41(tt, V2) → A__ISNATILIST(V2)
A__U51(tt, V2) → A__U52(a__isNatList(V2))
A__U51(tt, V2) → A__ISNATLIST(V2)
A__U61(tt, V2) → A__U62(a__isNatIList(V2))
A__U61(tt, V2) → A__ISNATILIST(V2)
A__U71(tt, L, N) → A__U72(a__isNat(N), L)
A__U71(tt, L, N) → A__ISNAT(N)
A__U72(tt, L) → A__LENGTH(mark(L))
A__U72(tt, L) → MARK(L)
A__U91(tt, IL, M, N) → A__U92(a__isNat(M), IL, M, N)
A__U91(tt, IL, M, N) → A__ISNAT(M)
A__U92(tt, IL, M, N) → A__U93(a__isNat(N), IL, M, N)
A__U92(tt, IL, M, N) → A__ISNAT(N)
A__U93(tt, IL, M, N) → MARK(N)
A__ISNAT(length(V1)) → A__U11(a__isNatList(V1))
A__ISNAT(length(V1)) → A__ISNATLIST(V1)
A__ISNAT(s(V1)) → A__U21(a__isNat(V1))
A__ISNAT(s(V1)) → A__ISNAT(V1)
A__ISNATILIST(V) → A__U31(a__isNatList(V))
A__ISNATILIST(V) → A__ISNATLIST(V)
A__ISNATILIST(cons(V1, V2)) → A__U41(a__isNat(V1), V2)
A__ISNATILIST(cons(V1, V2)) → A__ISNAT(V1)
A__ISNATLIST(cons(V1, V2)) → A__U51(a__isNat(V1), V2)
A__ISNATLIST(cons(V1, V2)) → A__ISNAT(V1)
A__ISNATLIST(take(V1, V2)) → A__U61(a__isNat(V1), V2)
A__ISNATLIST(take(V1, V2)) → A__ISNAT(V1)
A__LENGTH(cons(N, L)) → A__U71(a__isNatList(L), L, N)
A__LENGTH(cons(N, L)) → A__ISNATLIST(L)
A__TAKE(0, IL) → A__U81(a__isNatIList(IL))
A__TAKE(0, IL) → A__ISNATILIST(IL)
A__TAKE(s(M), cons(N, IL)) → A__U91(a__isNatIList(IL), IL, M, N)
A__TAKE(s(M), cons(N, IL)) → A__ISNATILIST(IL)
MARK(zeros) → A__ZEROS
MARK(U11(X)) → A__U11(mark(X))
MARK(U11(X)) → MARK(X)
MARK(U21(X)) → A__U21(mark(X))
MARK(U21(X)) → MARK(X)
MARK(U31(X)) → A__U31(mark(X))
MARK(U31(X)) → MARK(X)
MARK(U41(X1, X2)) → A__U41(mark(X1), X2)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → A__U42(mark(X))
MARK(U42(X)) → MARK(X)
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(U51(X1, X2)) → A__U51(mark(X1), X2)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U52(X)) → A__U52(mark(X))
MARK(U52(X)) → MARK(X)
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(U61(X1, X2)) → A__U61(mark(X1), X2)
MARK(U61(X1, X2)) → MARK(X1)
MARK(U62(X)) → A__U62(mark(X))
MARK(U62(X)) → MARK(X)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U72(X1, X2)) → A__U72(mark(X1), X2)
MARK(U72(X1, X2)) → MARK(X1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(length(X)) → A__LENGTH(mark(X))
MARK(length(X)) → MARK(X)
MARK(U81(X)) → A__U81(mark(X))
MARK(U81(X)) → MARK(X)
MARK(U91(X1, X2, X3, X4)) → A__U91(mark(X1), X2, X3, X4)
MARK(U91(X1, X2, X3, X4)) → MARK(X1)
MARK(U92(X1, X2, X3, X4)) → A__U92(mark(X1), X2, X3, X4)
MARK(U92(X1, X2, X3, X4)) → MARK(X1)
MARK(U93(X1, X2, X3, X4)) → A__U93(mark(X1), X2, X3, X4)
MARK(U93(X1, X2, X3, X4)) → MARK(X1)
MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)
MARK(cons(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt) → tt
a__U21(tt) → tt
a__U31(tt) → tt
a__U41(tt, V2) → a__U42(a__isNatIList(V2))
a__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatList(V2))
a__U52(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIList(V2))
a__U62(tt) → tt
a__U71(tt, L, N) → a__U72(a__isNat(N), L)
a__U72(tt, L) → s(a__length(mark(L)))
a__U81(tt) → nil
a__U91(tt, IL, M, N) → a__U92(a__isNat(M), IL, M, N)
a__U92(tt, IL, M, N) → a__U93(a__isNat(N), IL, M, N)
a__U93(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatList(V1))
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNatIList(V) → a__U31(a__isNatList(V))
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNat(V1), V2)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
a__isNatList(take(V1, V2)) → a__U61(a__isNat(V1), V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U71(a__isNatList(L), L, N)
a__take(0, IL) → a__U81(a__isNatIList(IL))
a__take(s(M), cons(N, IL)) → a__U91(a__isNatIList(IL), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X)) → a__U11(mark(X))
mark(U21(X)) → a__U21(mark(X))
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X)) → a__U62(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2)) → a__U72(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(U81(X)) → a__U81(mark(X))
mark(U91(X1, X2, X3, X4)) → a__U91(mark(X1), X2, X3, X4)
mark(U92(X1, X2, X3, X4)) → a__U92(mark(X1), X2, X3, X4)
mark(U93(X1, X2, X3, X4)) → a__U93(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X) → U11(X)
a__U21(X) → U21(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__isNatList(X) → isNatList(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2) → U72(X1, X2)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3, X4) → U91(X1, X2, X3, X4)
a__U92(X1, X2, X3, X4) → U92(X1, X2, X3, X4)
a__U93(X1, X2, X3, X4) → U93(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)

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

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 27 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

A__U41(tt, V2) → A__ISNATILIST(V2)
A__ISNATILIST(V) → A__ISNATLIST(V)
A__ISNATLIST(cons(V1, V2)) → A__U51(a__isNat(V1), V2)
A__U51(tt, V2) → A__ISNATLIST(V2)
A__ISNATLIST(cons(V1, V2)) → A__ISNAT(V1)
A__ISNAT(length(V1)) → A__ISNATLIST(V1)
A__ISNATLIST(take(V1, V2)) → A__U61(a__isNat(V1), V2)
A__U61(tt, V2) → A__ISNATILIST(V2)
A__ISNATILIST(cons(V1, V2)) → A__U41(a__isNat(V1), V2)
A__ISNATILIST(cons(V1, V2)) → A__ISNAT(V1)
A__ISNAT(s(V1)) → A__ISNAT(V1)
A__ISNATLIST(take(V1, V2)) → A__ISNAT(V1)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt) → tt
a__U21(tt) → tt
a__U31(tt) → tt
a__U41(tt, V2) → a__U42(a__isNatIList(V2))
a__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatList(V2))
a__U52(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIList(V2))
a__U62(tt) → tt
a__U71(tt, L, N) → a__U72(a__isNat(N), L)
a__U72(tt, L) → s(a__length(mark(L)))
a__U81(tt) → nil
a__U91(tt, IL, M, N) → a__U92(a__isNat(M), IL, M, N)
a__U92(tt, IL, M, N) → a__U93(a__isNat(N), IL, M, N)
a__U93(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatList(V1))
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNatIList(V) → a__U31(a__isNatList(V))
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNat(V1), V2)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
a__isNatList(take(V1, V2)) → a__U61(a__isNat(V1), V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U71(a__isNatList(L), L, N)
a__take(0, IL) → a__U81(a__isNatIList(IL))
a__take(s(M), cons(N, IL)) → a__U91(a__isNatIList(IL), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X)) → a__U11(mark(X))
mark(U21(X)) → a__U21(mark(X))
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X)) → a__U62(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2)) → a__U72(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(U81(X)) → a__U81(mark(X))
mark(U91(X1, X2, X3, X4)) → a__U91(mark(X1), X2, X3, X4)
mark(U92(X1, X2, X3, X4)) → a__U92(mark(X1), X2, X3, X4)
mark(U93(X1, X2, X3, X4)) → a__U93(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X) → U11(X)
a__U21(X) → U21(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__isNatList(X) → isNatList(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2) → U72(X1, X2)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3, X4) → U91(X1, X2, X3, X4)
a__U92(X1, X2, X3, X4) → U92(X1, X2, X3, X4)
a__U93(X1, X2, X3, X4) → U93(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)

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

(6) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

A__U41(tt, V2) → A__ISNATILIST(V2)
A__ISNATLIST(cons(V1, V2)) → A__U51(a__isNat(V1), V2)
A__U51(tt, V2) → A__ISNATLIST(V2)
A__ISNATLIST(cons(V1, V2)) → A__ISNAT(V1)
A__ISNAT(length(V1)) → A__ISNATLIST(V1)
A__ISNATLIST(take(V1, V2)) → A__U61(a__isNat(V1), V2)
A__U61(tt, V2) → A__ISNATILIST(V2)
A__ISNATILIST(cons(V1, V2)) → A__U41(a__isNat(V1), V2)
A__ISNATILIST(cons(V1, V2)) → A__ISNAT(V1)
A__ISNAT(s(V1)) → A__ISNAT(V1)
A__ISNATLIST(take(V1, V2)) → A__ISNAT(V1)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
A__U41(x1, x2) = A__U41(x2)
tt = tt
A__ISNATILIST(x1) = x1
A__ISNATLIST(x1) = x1
cons(x1, x2) = cons(x1, x2)
A__U51(x1, x2) = A__U51(x1, x2)
a__isNat(x1) = x1
A__ISNAT(x1) = x1
length(x1) = length(x1)
take(x1, x2) = take(x1, x2)
A__U61(x1, x2) = A__U61(x1, x2)
s(x1) = s(x1)
a__U61(x1, x2) = a__U61(x1, x2)
a__U62(x1) = a__U62(x1)
a__isNatIList(x1) = a__isNatIList(x1)
a__U52(x1) = a__U52
isNat(x1) = x1
a__U41(x1, x2) = a__U41
a__U42(x1) = a__U42
a__U31(x1) = a__U31
a__U51(x1, x2) = a__U51
a__isNatList(x1) = a__isNatList(x1)
0 = 0
a__U21(x1) = a__U21(x1)
a__U11(x1) = a__U11
U11(x1) = U11
U21(x1) = U21(x1)
U31(x1) = U31
U41(x1, x2) = U41
U42(x1) = U42
isNatIList(x1) = isNatIList
U51(x1, x2) = U51
U52(x1) = U52
isNatList(x1) = isNatList(x1)
U61(x1, x2) = U61(x1, x2)
U62(x1) = U62(x1)
nil = nil
zeros = zeros

Lexicographic path order with status [LPO].
Quasi-Precedence:

[A_{_U41₁}, cons₂] > A_{_U51₂} > [a_{_U62₁}, U62₁]
[A_{_U41₁}, cons₂] > [a_{_U41}, a_{_U42}, U42] > a_{_{isNatIList₁}} > [a_{_U31}, U31] > [tt, s₁, 0, a_{_U21₁}] > U21₁ > [a_{_U62₁}, U62₁]
[A_{_U41₁}, cons₂] > [a_{_U41}, a_{_U42}, U42] > a_{_{isNatIList₁}} > a_{_isNatList₁} > [a_{_U52}, a_{_U51}, U51, U52] > [tt, s₁, 0, a_{_U21₁}] > U21₁ > [a_{_U62₁}, U62₁]
[A_{_U41₁}, cons₂] > [a_{_U41}, a_{_U42}, U42] > a_{_{isNatIList₁}} > isNatIList > [a_{_U62₁}, U62₁]
[A_{_U41₁}, cons₂] > [a_{_U41}, a_{_U42}, U42] > U41 > [a_{_U62₁}, U62₁]
[length₁, a_{_U11}, U11] > a_{_isNatList₁} > [a_{_U52}, a_{_U51}, U51, U52] > [tt, s₁, 0, a_{_U21₁}] > U21₁ > [a_{_U62₁}, U62₁]
[take₂, a_{_U61₂}, U61₂] > A_{_U61₂} > [a_{_U62₁}, U62₁]
[take₂, a_{_U61₂}, U61₂] > a_{_{isNatIList₁}} > [a_{_U31}, U31] > [tt, s₁, 0, a_{_U21₁}] > U21₁ > [a_{_U62₁}, U62₁]
[take₂, a_{_U61₂}, U61₂] > a_{_{isNatIList₁}} > a_{_isNatList₁} > [a_{_U52}, a_{_U51}, U51, U52] > [tt, s₁, 0, a_{_U21₁}] > U21₁ > [a_{_U62₁}, U62₁]
[take₂, a_{_U61₂}, U61₂] > a_{_{isNatIList₁}} > isNatIList > [a_{_U62₁}, U62₁]
isNatList₁ > [a_{_U62₁}, U62₁]
nil > [tt, s₁, 0, a_{_U21₁}] > U21₁ > [a_{_U62₁}, U62₁]
zeros > [tt, s₁, 0, a_{_U21₁}] > U21₁ > [a_{_U62₁}, U62₁]

Status:

U31: []
U42: []
U11: []
a_{_{isNatIList₁}}: [1]
isNatIList: []
take₂: [1,2]
a_{_U41}: []
A_{_U61₂}: [2,1]
a_{_U21₁}: [1]
a_{_U52}: []
a_{_U31}: []
tt: []
isNatList₁: [1]
zeros: []
U41: []
A_{_U41₁}: [1]
s₁: [1]
a_{_isNatList₁}: [1]
nil: []
a_{_U62₁}: [1]
U51: []
a_{_U61₂}: [1,2]
a_{_U42}: []
a_{_U11}: []
U52: []
0: []
U62₁: [1]
cons₂: [2,1]
A_{_U51₂}: [2,1]
U61₂: [1,2]
a_{_U51}: []
length₁: [1]
U21₁: [1]

The following usable rules [FROCOS05] were oriented:

a__U61(tt, V2) → a__U62(a__isNatIList(V2))
a__U52(tt) → tt
a__isNat(X) → isNat(X)
a__U62(tt) → tt
a__U41(tt, V2) → a__U42(a__isNatIList(V2))
a__U31(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatList(V2))
a__U42(tt) → tt
a__isNat(0) → tt
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(length(V1)) → a__U11(a__isNatList(V1))
a__U11(X) → U11(X)
a__U21(X) → U21(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__U11(tt) → tt
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U21(tt) → tt
a__U52(X) → U52(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__isNatIList(cons(V1, V2)) → a__U41(a__isNat(V1), V2)
a__isNatIList(V) → a__U31(a__isNatList(V))
a__isNatIList(zeros) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
a__isNatList(take(V1, V2)) → a__U61(a__isNat(V1), V2)

(7) Obligation:

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

A__ISNATILIST(V) → A__ISNATLIST(V)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt) → tt
a__U21(tt) → tt
a__U31(tt) → tt
a__U41(tt, V2) → a__U42(a__isNatIList(V2))
a__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatList(V2))
a__U52(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIList(V2))
a__U62(tt) → tt
a__U71(tt, L, N) → a__U72(a__isNat(N), L)
a__U72(tt, L) → s(a__length(mark(L)))
a__U81(tt) → nil
a__U91(tt, IL, M, N) → a__U92(a__isNat(M), IL, M, N)
a__U92(tt, IL, M, N) → a__U93(a__isNat(N), IL, M, N)
a__U93(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatList(V1))
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNatIList(V) → a__U31(a__isNatList(V))
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNat(V1), V2)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
a__isNatList(take(V1, V2)) → a__U61(a__isNat(V1), V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U71(a__isNatList(L), L, N)
a__take(0, IL) → a__U81(a__isNatIList(IL))
a__take(s(M), cons(N, IL)) → a__U91(a__isNatIList(IL), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X)) → a__U11(mark(X))
mark(U21(X)) → a__U21(mark(X))
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X)) → a__U62(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2)) → a__U72(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(U81(X)) → a__U81(mark(X))
mark(U91(X1, X2, X3, X4)) → a__U91(mark(X1), X2, X3, X4)
mark(U92(X1, X2, X3, X4)) → a__U92(mark(X1), X2, X3, X4)
mark(U93(X1, X2, X3, X4)) → a__U93(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X) → U11(X)
a__U21(X) → U21(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__isNatList(X) → isNatList(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2) → U72(X1, X2)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3, X4) → U91(X1, X2, X3, X4)
a__U92(X1, X2, X3, X4) → U92(X1, X2, X3, X4)
a__U93(X1, X2, X3, X4) → U93(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)

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

(8) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(9) TRUE

(10) Obligation:

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

MARK(U11(X)) → MARK(X)
MARK(U21(X)) → MARK(X)
MARK(U31(X)) → MARK(X)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U42(X)) → MARK(X)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U52(X)) → MARK(X)
MARK(U61(X1, X2)) → MARK(X1)
MARK(U62(X)) → MARK(X)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
A__U71(tt, L, N) → A__U72(a__isNat(N), L)
A__U72(tt, L) → A__LENGTH(mark(L))
A__LENGTH(cons(N, L)) → A__U71(a__isNatList(L), L, N)
A__U72(tt, L) → MARK(L)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U72(X1, X2)) → A__U72(mark(X1), X2)
MARK(U72(X1, X2)) → MARK(X1)
MARK(length(X)) → A__LENGTH(mark(X))
MARK(length(X)) → MARK(X)
MARK(U81(X)) → MARK(X)
MARK(U91(X1, X2, X3, X4)) → A__U91(mark(X1), X2, X3, X4)
A__U91(tt, IL, M, N) → A__U92(a__isNat(M), IL, M, N)
A__U92(tt, IL, M, N) → A__U93(a__isNat(N), IL, M, N)
A__U93(tt, IL, M, N) → MARK(N)
MARK(U91(X1, X2, X3, X4)) → MARK(X1)
MARK(U92(X1, X2, X3, X4)) → A__U92(mark(X1), X2, X3, X4)
MARK(U92(X1, X2, X3, X4)) → MARK(X1)
MARK(U93(X1, X2, X3, X4)) → A__U93(mark(X1), X2, X3, X4)
MARK(U93(X1, X2, X3, X4)) → MARK(X1)
MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
A__TAKE(s(M), cons(N, IL)) → A__U91(a__isNatIList(IL), IL, M, N)
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)
MARK(cons(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)

The TRS R consists of the following rules:

a__zeros → cons(0, zeros)
a__U11(tt) → tt
a__U21(tt) → tt
a__U31(tt) → tt
a__U41(tt, V2) → a__U42(a__isNatIList(V2))
a__U42(tt) → tt
a__U51(tt, V2) → a__U52(a__isNatList(V2))
a__U52(tt) → tt
a__U61(tt, V2) → a__U62(a__isNatIList(V2))
a__U62(tt) → tt
a__U71(tt, L, N) → a__U72(a__isNat(N), L)
a__U72(tt, L) → s(a__length(mark(L)))
a__U81(tt) → nil
a__U91(tt, IL, M, N) → a__U92(a__isNat(M), IL, M, N)
a__U92(tt, IL, M, N) → a__U93(a__isNat(N), IL, M, N)
a__U93(tt, IL, M, N) → cons(mark(N), take(M, IL))
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatList(V1))
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNatIList(V) → a__U31(a__isNatList(V))
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__isNat(V1), V2)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
a__isNatList(take(V1, V2)) → a__U61(a__isNat(V1), V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U71(a__isNatList(L), L, N)
a__take(0, IL) → a__U81(a__isNatIList(IL))
a__take(s(M), cons(N, IL)) → a__U91(a__isNatIList(IL), IL, M, N)
mark(zeros) → a__zeros
mark(U11(X)) → a__U11(mark(X))
mark(U21(X)) → a__U21(mark(X))
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U42(X)) → a__U42(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U62(X)) → a__U62(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2)) → a__U72(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
mark(U81(X)) → a__U81(mark(X))
mark(U91(X1, X2, X3, X4)) → a__U91(mark(X1), X2, X3, X4)
mark(U92(X1, X2, X3, X4)) → a__U92(mark(X1), X2, X3, X4)
mark(U93(X1, X2, X3, X4)) → a__U93(mark(X1), X2, X3, X4)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X) → U11(X)
a__U21(X) → U21(X)
a__U31(X) → U31(X)
a__U41(X1, X2) → U41(X1, X2)
a__U42(X) → U42(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2) → U51(X1, X2)
a__U52(X) → U52(X)
a__isNatList(X) → isNatList(X)
a__U61(X1, X2) → U61(X1, X2)
a__U62(X) → U62(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2) → U72(X1, X2)
a__isNat(X) → isNat(X)
a__length(X) → length(X)
a__U81(X) → U81(X)
a__U91(X1, X2, X3, X4) → U91(X1, X2, X3, X4)
a__U92(X1, X2, X3, X4) → U92(X1, X2, X3, X4)
a__U93(X1, X2, X3, X4) → U93(X1, X2, X3, X4)
a__take(X1, X2) → take(X1, X2)

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