Termination w.r.t. Q of the given QTRS could be proven:

0 QTRS
↳1 QTRSRRRProof (⇔)
↳2 QTRS
↳3 QTRSRRRProof (⇔)
↳4 QTRS
↳5 QTRSRRRProof (⇔)
↳6 QTRS
↳7 QTRSRRRProof (⇔)
↳8 QTRS
↳9 QTRSRRRProof (⇔)
↳10 QTRS
↳11 QTRSRRRProof (⇔)
↳12 QTRS
↳13 QTRSRRRProof (⇔)
↳14 QTRS
↳15 QTRSRRRProof (⇔)
↳16 QTRS
↳17 QTRSRRRProof (⇔)
↳18 QTRS
↳19 QTRSRRRProof (⇔)
↳20 QTRS
  ↳21 RisEmptyProof (⇔)
    ↳22 TRUE
  ↳23 RisEmptyProof (⇔)
    ↳24 TRUE

(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, L, N) → a__U62(a__isNat(N), L)
a__U62(tt, L) → s(a__length(mark(L)))
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__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__isNatList(L), L, 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, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
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, X3) → U61(X1, X2, X3)
a__U62(X1, X2) → U62(X1, X2)
a__isNat(X) → isNat(X)
a__length(X) → length(X)

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0   
POL(U11(x1)) = x1   
POL(U21(x1)) = x1   
POL(U31(x1)) = x1   
POL(U41(x1, x2)) = x1 + x2   
POL(U42(x1)) = x1   
POL(U51(x1, x2)) = x1 + x2   
POL(U52(x1)) = x1   
POL(U61(x1, x2, x3)) = x1 + 2·x2 + x3   
POL(U62(x1, x2)) = x1 + 2·x2   
POL(a__U11(x1)) = x1   
POL(a__U21(x1)) = x1   
POL(a__U31(x1)) = x1   
POL(a__U41(x1, x2)) = x1 + x2   
POL(a__U42(x1)) = x1   
POL(a__U51(x1, x2)) = x1 + x2   
POL(a__U52(x1)) = x1   
POL(a__U61(x1, x2, x3)) = x1 + 2·x2 + x3   
POL(a__U62(x1, x2)) = x1 + 2·x2   
POL(a__isNat(x1)) = x1   
POL(a__isNatIList(x1)) = x1   
POL(a__isNatList(x1)) = x1   
POL(a__length(x1)) = 2·x1   
POL(a__zeros) = 0   
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(mark(x1)) = x1   
POL(nil) = 1   
POL(s(x1)) = x1   
POL(tt) = 0   
POL(zeros) = 0   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

a__isNatList(nil) → tt
a__length(nil) → 0


(2) 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, L, N) → a__U62(a__isNat(N), L)
a__U62(tt, L) → s(a__length(mark(L)))
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(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
a__length(cons(N, L)) → a__U61(a__isNatList(L), L, 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, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
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, X3) → U61(X1, X2, X3)
a__U62(X1, X2) → U62(X1, X2)
a__isNat(X) → isNat(X)
a__length(X) → length(X)

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0   
POL(U11(x1)) = x1   
POL(U21(x1)) = x1   
POL(U31(x1)) = x1   
POL(U41(x1, x2)) = 1 + x1 + x2   
POL(U42(x1)) = x1   
POL(U51(x1, x2)) = x1 + x2   
POL(U52(x1)) = x1   
POL(U61(x1, x2, x3)) = x1 + 2·x2 + 2·x3   
POL(U62(x1, x2)) = x1 + 2·x2   
POL(a__U11(x1)) = x1   
POL(a__U21(x1)) = x1   
POL(a__U31(x1)) = x1   
POL(a__U41(x1, x2)) = 1 + x1 + x2   
POL(a__U42(x1)) = x1   
POL(a__U51(x1, x2)) = x1 + x2   
POL(a__U52(x1)) = x1   
POL(a__U61(x1, x2, x3)) = x1 + 2·x2 + 2·x3   
POL(a__U62(x1, x2)) = x1 + 2·x2   
POL(a__isNat(x1)) = 2·x1   
POL(a__isNatIList(x1)) = 1 + x1   
POL(a__isNatList(x1)) = x1   
POL(a__length(x1)) = 2·x1   
POL(a__zeros) = 0   
POL(cons(x1, x2)) = 2·x1 + 2·x2   
POL(isNat(x1)) = 2·x1   
POL(isNatIList(x1)) = 1 + x1   
POL(isNatList(x1)) = x1   
POL(length(x1)) = 2·x1   
POL(mark(x1)) = x1   
POL(nil) = 0   
POL(s(x1)) = x1   
POL(tt) = 0   
POL(zeros) = 0   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

a__isNatIList(V) → a__U31(a__isNatList(V))
a__isNatIList(zeros) → tt


(4) 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, L, N) → a__U62(a__isNat(N), L)
a__U62(tt, L) → s(a__length(mark(L)))
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatList(V1))
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNatIList(cons(V1, V2)) → a__U41(a__isNat(V1), V2)
a__isNatList(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
a__length(cons(N, L)) → a__U61(a__isNatList(L), L, 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, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
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, X3) → U61(X1, X2, X3)
a__U62(X1, X2) → U62(X1, X2)
a__isNat(X) → isNat(X)
a__length(X) → length(X)

Q is empty.

(5) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0   
POL(U11(x1)) = 2·x1   
POL(U21(x1)) = x1   
POL(U31(x1)) = 1 + 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 + x3   
POL(U62(x1, x2)) = x1 + 2·x2   
POL(a__U11(x1)) = 2·x1   
POL(a__U21(x1)) = x1   
POL(a__U31(x1)) = 1 + x1   
POL(a__U41(x1, x2)) = 2·x1 + 2·x2   
POL(a__U42(x1)) = 2·x1   
POL(a__U51(x1, x2)) = x1 + 2·x2   
POL(a__U52(x1)) = x1   
POL(a__U61(x1, x2, x3)) = 2·x1 + 2·x2 + x3   
POL(a__U62(x1, x2)) = x1 + 2·x2   
POL(a__isNat(x1)) = x1   
POL(a__isNatIList(x1)) = x1   
POL(a__isNatList(x1)) = x1   
POL(a__length(x1)) = 2·x1   
POL(a__zeros) = 0   
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(mark(x1)) = x1   
POL(nil) = 0   
POL(s(x1)) = x1   
POL(tt) = 0   
POL(zeros) = 0   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

a__U31(tt) → tt


(6) 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__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, L, N) → a__U62(a__isNat(N), L)
a__U62(tt, L) → s(a__length(mark(L)))
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatList(V1))
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNatIList(cons(V1, V2)) → a__U41(a__isNat(V1), V2)
a__isNatList(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
a__length(cons(N, L)) → a__U61(a__isNatList(L), L, 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, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
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, X3) → U61(X1, X2, X3)
a__U62(X1, X2) → U62(X1, X2)
a__isNat(X) → isNat(X)
a__length(X) → length(X)

Q is empty.

(7) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0   
POL(U11(x1)) = x1   
POL(U21(x1)) = x1   
POL(U31(x1)) = x1   
POL(U41(x1, x2)) = x1 + 2·x2   
POL(U42(x1)) = x1   
POL(U51(x1, x2)) = x1 + x2   
POL(U52(x1)) = x1   
POL(U61(x1, x2, x3)) = 1 + 2·x1 + 2·x2 + 2·x3   
POL(U62(x1, x2)) = 1 + x1 + 2·x2   
POL(a__U11(x1)) = x1   
POL(a__U21(x1)) = x1   
POL(a__U31(x1)) = x1   
POL(a__U41(x1, x2)) = x1 + 2·x2   
POL(a__U42(x1)) = x1   
POL(a__U51(x1, x2)) = x1 + x2   
POL(a__U52(x1)) = x1   
POL(a__U61(x1, x2, x3)) = 1 + 2·x1 + 2·x2 + 2·x3   
POL(a__U62(x1, x2)) = 1 + x1 + 2·x2   
POL(a__isNat(x1)) = x1   
POL(a__isNatIList(x1)) = 2·x1   
POL(a__isNatList(x1)) = x1   
POL(a__length(x1)) = 1 + 2·x1   
POL(a__zeros) = 0   
POL(cons(x1, x2)) = 2·x1 + 2·x2   
POL(isNat(x1)) = x1   
POL(isNatIList(x1)) = 2·x1   
POL(isNatList(x1)) = x1   
POL(length(x1)) = 1 + 2·x1   
POL(mark(x1)) = x1   
POL(nil) = 0   
POL(s(x1)) = x1   
POL(tt) = 0   
POL(zeros) = 0   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

a__isNat(length(V1)) → a__U11(a__isNatList(V1))


(8) 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__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, L, N) → a__U62(a__isNat(N), L)
a__U62(tt, L) → s(a__length(mark(L)))
a__isNat(0) → tt
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNatIList(cons(V1, V2)) → a__U41(a__isNat(V1), V2)
a__isNatList(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
a__length(cons(N, L)) → a__U61(a__isNatList(L), L, 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, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
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, X3) → U61(X1, X2, X3)
a__U62(X1, X2) → U62(X1, X2)
a__isNat(X) → isNat(X)
a__length(X) → length(X)

Q is empty.

(9) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0   
POL(U11(x1)) = 1 + x1   
POL(U21(x1)) = x1   
POL(U31(x1)) = 2·x1   
POL(U41(x1, x2)) = 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(a__U11(x1)) = 1 + x1   
POL(a__U21(x1)) = x1   
POL(a__U31(x1)) = 2·x1   
POL(a__U41(x1, x2)) = x1 + 2·x2   
POL(a__U42(x1)) = x1   
POL(a__U51(x1, x2)) = 2·x1 + 2·x2   
POL(a__U52(x1)) = x1   
POL(a__U61(x1, x2, x3)) = x1 + 2·x2 + 2·x3   
POL(a__U62(x1, x2)) = 2·x1 + 2·x2   
POL(a__isNat(x1)) = x1   
POL(a__isNatIList(x1)) = 2·x1   
POL(a__isNatList(x1)) = 2·x1   
POL(a__length(x1)) = 2·x1   
POL(a__zeros) = 0   
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(mark(x1)) = x1   
POL(nil) = 0   
POL(s(x1)) = x1   
POL(tt) = 0   
POL(zeros) = 0   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

a__U11(tt) → tt


(10) Obligation:

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

a__zeros → cons(0, zeros)
a__U21(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, L, N) → a__U62(a__isNat(N), L)
a__U62(tt, L) → s(a__length(mark(L)))
a__isNat(0) → tt
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNatIList(cons(V1, V2)) → a__U41(a__isNat(V1), V2)
a__isNatList(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
a__length(cons(N, L)) → a__U61(a__isNatList(L), L, 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, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(length(X)) → a__length(mark(X))
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, X3) → U61(X1, X2, X3)
a__U62(X1, X2) → U62(X1, X2)
a__isNat(X) → isNat(X)
a__length(X) → length(X)

Q is empty.

(11) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 1   
POL(U11(x1)) = x1   
POL(U21(x1)) = x1   
POL(U31(x1)) = x1   
POL(U41(x1, x2)) = 1 + x1 + x2   
POL(U42(x1)) = x1   
POL(U51(x1, x2)) = x1 + x2   
POL(U52(x1)) = x1   
POL(U61(x1, x2, x3)) = x1 + x2 + x3   
POL(U62(x1, x2)) = x1 + x2   
POL(a__U11(x1)) = x1   
POL(a__U21(x1)) = x1   
POL(a__U31(x1)) = x1   
POL(a__U41(x1, x2)) = 1 + x1 + x2   
POL(a__U42(x1)) = x1   
POL(a__U51(x1, x2)) = x1 + x2   
POL(a__U52(x1)) = x1   
POL(a__U61(x1, x2, x3)) = x1 + x2 + x3   
POL(a__U62(x1, x2)) = x1 + x2   
POL(a__isNat(x1)) = x1   
POL(a__isNatIList(x1)) = 2 + x1   
POL(a__isNatList(x1)) = x1   
POL(a__length(x1)) = x1   
POL(a__zeros) = 1   
POL(cons(x1, x2)) = x1 + 2·x2   
POL(isNat(x1)) = x1   
POL(isNatIList(x1)) = 2 + x1   
POL(isNatList(x1)) = x1   
POL(length(x1)) = x1   
POL(mark(x1)) = 1 + x1   
POL(nil) = 0   
POL(s(x1)) = x1   
POL(tt) = 1   
POL(zeros) = 0   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

a__U51(tt, V2) → a__U52(a__isNatList(V2))
a__U61(tt, L, N) → a__U62(a__isNat(N), L)
a__isNatIList(cons(V1, V2)) → a__U41(a__isNat(V1), V2)
mark(isNatIList(X)) → a__isNatIList(X)
mark(isNatList(X)) → a__isNatList(X)
mark(isNat(X)) → a__isNat(X)
mark(0) → 0
mark(tt) → tt
mark(nil) → nil
a__zeros → zeros


(12) Obligation:

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

a__zeros → cons(0, zeros)
a__U21(tt) → tt
a__U41(tt, V2) → a__U42(a__isNatIList(V2))
a__U42(tt) → tt
a__U52(tt) → tt
a__U62(tt, L) → s(a__length(mark(L)))
a__isNat(0) → tt
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNatList(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
a__length(cons(N, L)) → a__U61(a__isNatList(L), L, 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(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
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, X3) → U61(X1, X2, X3)
a__U62(X1, X2) → U62(X1, X2)
a__isNat(X) → isNat(X)
a__length(X) → length(X)

Q is empty.

(13) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0   
POL(U11(x1)) = 2·x1   
POL(U21(x1)) = x1   
POL(U31(x1)) = 2·x1   
POL(U41(x1, x2)) = 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 + 2·x3   
POL(U62(x1, x2)) = 2 + x1 + 2·x2   
POL(a__U11(x1)) = 2·x1   
POL(a__U21(x1)) = x1   
POL(a__U31(x1)) = 2·x1   
POL(a__U41(x1, x2)) = x1 + 2·x2   
POL(a__U42(x1)) = x1   
POL(a__U51(x1, x2)) = x1 + 2·x2   
POL(a__U52(x1)) = x1   
POL(a__U61(x1, x2, x3)) = x1 + 2·x2 + 2·x3   
POL(a__U62(x1, x2)) = 2 + x1 + 2·x2   
POL(a__isNat(x1)) = 2·x1   
POL(a__isNatIList(x1)) = x1   
POL(a__isNatList(x1)) = 2·x1   
POL(a__length(x1)) = 2·x1   
POL(a__zeros) = 0   
POL(cons(x1, x2)) = 2·x1 + 2·x2   
POL(isNat(x1)) = 2·x1   
POL(isNatIList(x1)) = x1   
POL(isNatList(x1)) = 2·x1   
POL(length(x1)) = 2·x1   
POL(mark(x1)) = x1   
POL(s(x1)) = x1   
POL(tt) = 0   
POL(zeros) = 0   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

a__U62(tt, L) → s(a__length(mark(L)))


(14) Obligation:

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

a__zeros → cons(0, zeros)
a__U21(tt) → tt
a__U41(tt, V2) → a__U42(a__isNatIList(V2))
a__U42(tt) → tt
a__U52(tt) → tt
a__isNat(0) → tt
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNatList(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
a__length(cons(N, L)) → a__U61(a__isNatList(L), L, 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(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
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, X3) → U61(X1, X2, X3)
a__U62(X1, X2) → U62(X1, X2)
a__isNat(X) → isNat(X)
a__length(X) → length(X)

Q is empty.

(15) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0   
POL(U11(x1)) = 1 + x1   
POL(U21(x1)) = 1 + x1   
POL(U31(x1)) = 1 + x1   
POL(U41(x1, x2)) = 1 + x1 + x2   
POL(U42(x1)) = 1 + x1   
POL(U51(x1, x2)) = 1 + x1 + 2·x2   
POL(U52(x1)) = 1 + x1   
POL(U61(x1, x2, x3)) = 1 + x1 + x2 + x3   
POL(U62(x1, x2)) = 1 + x1 + x2   
POL(a__U11(x1)) = 2 + x1   
POL(a__U21(x1)) = 2 + x1   
POL(a__U31(x1)) = 2 + x1   
POL(a__U41(x1, x2)) = 2 + x1 + 2·x2   
POL(a__U42(x1)) = 2 + x1   
POL(a__U51(x1, x2)) = 2 + x1 + 2·x2   
POL(a__U52(x1)) = 2 + x1   
POL(a__U61(x1, x2, x3)) = 2 + x1 + x2 + x3   
POL(a__U62(x1, x2)) = 2 + x1 + 2·x2   
POL(a__isNat(x1)) = 2 + 2·x1   
POL(a__isNatIList(x1)) = 2 + 2·x1   
POL(a__isNatList(x1)) = 2 + x1   
POL(a__length(x1)) = 2 + x1   
POL(a__zeros) = 2   
POL(cons(x1, x2)) = 2 + 2·x1 + 2·x2   
POL(isNat(x1)) = 1 + x1   
POL(isNatIList(x1)) = 1 + x1   
POL(isNatList(x1)) = 1 + x1   
POL(length(x1)) = 1 + x1   
POL(mark(x1)) = 2 + 2·x1   
POL(s(x1)) = 1 + x1   
POL(tt) = 2   
POL(zeros) = 0   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

a__U21(tt) → tt
a__U42(tt) → tt
a__U52(tt) → tt
mark(s(X)) → s(mark(X))
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, X3) → U61(X1, X2, X3)
a__U62(X1, X2) → U62(X1, X2)
a__isNat(X) → isNat(X)
a__length(X) → length(X)


(16) Obligation:

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

a__zeros → cons(0, zeros)
a__U41(tt, V2) → a__U42(a__isNatIList(V2))
a__isNat(0) → tt
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNatList(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
a__length(cons(N, L)) → a__U61(a__isNatList(L), L, 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(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)

Q is empty.

(17) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0   
POL(U11(x1)) = 2 + x1   
POL(U21(x1)) = 2 + x1   
POL(U31(x1)) = 2 + 2·x1   
POL(U41(x1, x2)) = 1 + 2·x1 + 2·x2   
POL(U42(x1)) = 2 + 2·x1   
POL(U51(x1, x2)) = 1 + x1 + x2   
POL(U52(x1)) = 2 + 2·x1   
POL(U61(x1, x2, x3)) = 2 + x1 + 2·x2 + 2·x3   
POL(U62(x1, x2)) = 1 + 2·x1 + 2·x2   
POL(a__U11(x1)) = 2 + x1   
POL(a__U21(x1)) = x1   
POL(a__U31(x1)) = 2 + 2·x1   
POL(a__U41(x1, x2)) = x1 + 2·x2   
POL(a__U42(x1)) = x1   
POL(a__U51(x1, x2)) = x1 + 2·x2   
POL(a__U52(x1)) = 1 + x1   
POL(a__U61(x1, x2, x3)) = x1 + x2 + x3   
POL(a__U62(x1, x2)) = x1 + x2   
POL(a__isNat(x1)) = x1   
POL(a__isNatIList(x1)) = 2·x1   
POL(a__isNatList(x1)) = x1   
POL(a__length(x1)) = x1   
POL(a__zeros) = 1   
POL(cons(x1, x2)) = x1 + 2·x2   
POL(length(x1)) = 2·x1   
POL(mark(x1)) = 1 + 2·x1   
POL(s(x1)) = 1 + 2·x1   
POL(tt) = 0   
POL(zeros) = 0   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

a__zeros → cons(0, zeros)
a__isNat(s(V1)) → a__U21(a__isNat(V1))
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(U51(X1, X2)) → a__U51(mark(X1), X2)
mark(U52(X)) → a__U52(mark(X))
mark(U61(X1, X2, X3)) → a__U61(mark(X1), X2, X3)
mark(U62(X1, X2)) → a__U62(mark(X1), X2)


(18) Obligation:

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

a__U41(tt, V2) → a__U42(a__isNatIList(V2))
a__isNat(0) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
a__length(cons(N, L)) → a__U61(a__isNatList(L), L, N)
mark(zeros) → a__zeros
mark(length(X)) → a__length(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)

Q is empty.

(19) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 1   
POL(a__U41(x1, x2)) = 2 + 2·x1 + 2·x2   
POL(a__U42(x1)) = x1   
POL(a__U51(x1, x2)) = 2 + x1 + 2·x2   
POL(a__U61(x1, x2, x3)) = 1 + x1 + 2·x2 + 2·x3   
POL(a__isNat(x1)) = x1   
POL(a__isNatIList(x1)) = x1   
POL(a__isNatList(x1)) = 2·x1   
POL(a__length(x1)) = 2·x1   
POL(a__zeros) = 0   
POL(cons(x1, x2)) = 2 + x1 + 2·x2   
POL(length(x1)) = 2 + 2·x1   
POL(mark(x1)) = 1 + 2·x1   
POL(tt) = 0   
POL(zeros) = 0   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

a__U41(tt, V2) → a__U42(a__isNatIList(V2))
a__isNat(0) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__isNat(V1), V2)
a__length(cons(N, L)) → a__U61(a__isNatList(L), L, N)
mark(zeros) → a__zeros
mark(length(X)) → a__length(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)


(20) Obligation:

Q restricted rewrite system:
R is empty.
Q is empty.

(21) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.

(22) TRUE

(23) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.

(24) TRUE