Termination w.r.t. Q proof of /tmp/tmp_TPsFn/LengthOfFiniteLists_nokinds_noand

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

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.

↳ QTRS

  ↳ RRRPoloQTRSProof

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(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:

active(isNatList(nil)) → mark(tt)
active(length(nil)) → mark(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)) = x₁ + 2·x₂
POL(U42(x₁)) = 2·x₁
POL(U51(x₁, x₂)) = x₁ + x₂
POL(U52(x₁)) = x₁
POL(U61(x₁, x₂, x₃)) = 2·x₁ + 2·x₂ + 2·x₃
POL(U62(x₁, x₂)) = x₁ + 2·x₂
POL(active(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(mark(x₁)) = x₁
POL(nil) = 1
POL(ok(x₁)) = x₁
POL(proper(x₁)) = x₁
POL(s(x₁)) = x₁
POL(top(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(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:

active(U11(tt)) → mark(tt)

Used ordering:
Polynomial interpretation [25]:

POL(0) = 0
POL(U11(x₁)) = 1 + 2·x₁
POL(U21(x₁)) = x₁
POL(U31(x₁)) = x₁
POL(U41(x₁, x₂)) = x₁ + x₂
POL(U42(x₁)) = x₁
POL(U51(x₁, x₂)) = x₁ + 2·x₂
POL(U52(x₁)) = 2·x₁
POL(U61(x₁, x₂, x₃)) = 1 + 2·x₁ + 2·x₂ + 2·x₃
POL(U62(x₁, x₂)) = 1 + 2·x₁ + 2·x₂
POL(active(x₁)) = x₁
POL(cons(x₁, x₂)) = x₁ + 2·x₂
POL(isNat(x₁)) = x₁
POL(isNatIList(x₁)) = x₁
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = 1 + 2·x₁
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = x₁
POL(s(x₁)) = x₁
POL(top(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:

active(zeros) → mark(cons(0, zeros))
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.

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

active(zeros) → mark(cons(0, zeros))
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(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:

active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)

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 + 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 + 2·x₃
POL(U62(x₁, x₂)) = 2·x₁ + 2·x₂
POL(active(x₁)) = x₁
POL(cons(x₁, x₂)) = x₁ + 2·x₂
POL(isNat(x₁)) = x₁
POL(isNatIList(x₁)) = 2 + 2·x₁
POL(isNatList(x₁)) = 2·x₁
POL(length(x₁)) = 2·x₁
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = x₁
POL(s(x₁)) = x₁
POL(top(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:

active(zeros) → mark(cons(0, zeros))
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.

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

active(zeros) → mark(cons(0, zeros))
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(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:

active(U31(tt)) → mark(tt)

Used ordering:
Polynomial interpretation [25]:

POL(0) = 0
POL(U11(x₁)) = x₁
POL(U21(x₁)) = x₁
POL(U31(x₁)) = 1 + 2·x₁
POL(U41(x₁, x₂)) = 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 + 2·x₃
POL(U62(x₁, x₂)) = 2·x₁ + 2·x₂
POL(active(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(mark(x₁)) = x₁
POL(nil) = 2
POL(ok(x₁)) = x₁
POL(proper(x₁)) = x₁
POL(s(x₁)) = x₁
POL(top(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

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

active(zeros) → mark(cons(0, zeros))
active(U21(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.

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

active(zeros) → mark(cons(0, zeros))
active(U21(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(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:

active(isNat(length(V1))) → mark(U11(isNatList(V1)))

Used ordering:
Polynomial interpretation [25]:

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)) = 2·x₁
POL(U51(x₁, x₂)) = x₁ + 2·x₂
POL(U52(x₁)) = x₁
POL(U61(x₁, x₂, x₃)) = 2 + x₁ + x₂ + 2·x₃
POL(U62(x₁, x₂)) = 2 + x₁ + x₂
POL(active(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(mark(x₁)) = x₁
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = x₁
POL(s(x₁)) = x₁
POL(top(x₁)) = 2·x₁
POL(tt) = 0
POL(zeros) = 0

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ DependencyPairsProof

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

active(zeros) → mark(cons(0, zeros))
active(U21(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(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:

ACTIVE(U51(X1, X2)) → ACTIVE(X1)
ACTIVE(U41(tt, V2)) → U42¹(isNatIList(V2))
PROPER(U51(X1, X2)) → PROPER(X2)
ACTIVE(U51(tt, V2)) → U52¹(isNatList(V2))
ACTIVE(U31(X)) → U31¹(active(X))
PROPER(U42(X)) → U42¹(proper(X))
ACTIVE(cons(X1, X2)) → CONS(active(X1), X2)
PROPER(U62(X1, X2)) → PROPER(X2)
ACTIVE(U61(tt, L, N)) → U62¹(isNat(N), L)
PROPER(isNatList(X)) → PROPER(X)
PROPER(length(X)) → LENGTH(proper(X))
ACTIVE(U52(X)) → U52¹(active(X))
U41¹(ok(X1), ok(X2)) → U41¹(X1, X2)
ACTIVE(U62(tt, L)) → S(length(L))
ACTIVE(isNatList(cons(V1, V2))) → U51¹(isNat(V1), V2)
PROPER(length(X)) → PROPER(X)
U62¹(ok(X1), ok(X2)) → U62¹(X1, X2)
PROPER(U61(X1, X2, X3)) → PROPER(X3)
ACTIVE(U41(X1, X2)) → U41¹(active(X1), X2)
PROPER(isNatList(X)) → ISNATLIST(proper(X))
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(U31(X)) → ACTIVE(X)
S(ok(X)) → S(X)
CONS(mark(X1), X2) → CONS(X1, X2)
TOP(mark(X)) → PROPER(X)
ACTIVE(U42(X)) → ACTIVE(X)
ACTIVE(isNatIList(cons(V1, V2))) → U41¹(isNat(V1), V2)
PROPER(U11(X)) → U11¹(proper(X))
TOP(ok(X)) → ACTIVE(X)
ACTIVE(length(X)) → ACTIVE(X)
ACTIVE(isNat(s(V1))) → U21¹(isNat(V1))
PROPER(cons(X1, X2)) → PROPER(X1)
ACTIVE(U41(X1, X2)) → ACTIVE(X1)
ACTIVE(U51(tt, V2)) → ISNATLIST(V2)
PROPER(U21(X)) → U21¹(proper(X))
LENGTH(mark(X)) → LENGTH(X)
ACTIVE(U62(tt, L)) → LENGTH(L)
U51¹(mark(X1), X2) → U51¹(X1, X2)
S(mark(X)) → S(X)
PROPER(cons(X1, X2)) → CONS(proper(X1), proper(X2))
PROPER(U41(X1, X2)) → PROPER(X2)
ACTIVE(length(X)) → LENGTH(active(X))
ISNATLIST(ok(X)) → ISNATLIST(X)
U11¹(mark(X)) → U11¹(X)
ACTIVE(U51(X1, X2)) → U51¹(active(X1), X2)
PROPER(U61(X1, X2, X3)) → PROPER(X2)
ACTIVE(isNat(s(V1))) → ISNAT(V1)
U61¹(mark(X1), X2, X3) → U61¹(X1, X2, X3)
PROPER(U51(X1, X2)) → PROPER(X1)
PROPER(U42(X)) → PROPER(X)
PROPER(U21(X)) → PROPER(X)
U51¹(ok(X1), ok(X2)) → U51¹(X1, X2)
U31¹(mark(X)) → U31¹(X)
PROPER(U62(X1, X2)) → PROPER(X1)
ACTIVE(U62(X1, X2)) → ACTIVE(X1)
ACTIVE(U11(X)) → U11¹(active(X))
PROPER(U61(X1, X2, X3)) → U61¹(proper(X1), proper(X2), proper(X3))
U31¹(ok(X)) → U31¹(X)
U41¹(mark(X1), X2) → U41¹(X1, X2)
U61¹(ok(X1), ok(X2), ok(X3)) → U61¹(X1, X2, X3)
U42¹(ok(X)) → U42¹(X)
U21¹(ok(X)) → U21¹(X)
ACTIVE(zeros) → CONS(0, zeros)
PROPER(cons(X1, X2)) → PROPER(X2)
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(isNatIList(cons(V1, V2))) → ISNAT(V1)
ISNATILIST(ok(X)) → ISNATILIST(X)
ACTIVE(U61(tt, L, N)) → ISNAT(N)
PROPER(U11(X)) → PROPER(X)
ACTIVE(U42(X)) → U42¹(active(X))
U21¹(mark(X)) → U21¹(X)
PROPER(s(X)) → S(proper(X))
ACTIVE(isNatList(cons(V1, V2))) → ISNAT(V1)
PROPER(U31(X)) → PROPER(X)
PROPER(isNatIList(X)) → ISNATILIST(proper(X))
PROPER(U52(X)) → PROPER(X)
ACTIVE(U62(X1, X2)) → U62¹(active(X1), X2)
ACTIVE(U52(X)) → ACTIVE(X)
U52¹(mark(X)) → U52¹(X)
ACTIVE(length(cons(N, L))) → U61¹(isNatList(L), L, N)
PROPER(isNat(X)) → PROPER(X)
LENGTH(ok(X)) → LENGTH(X)
PROPER(isNatIList(X)) → PROPER(X)
PROPER(U41(X1, X2)) → U41¹(proper(X1), proper(X2))
PROPER(s(X)) → PROPER(X)
U62¹(mark(X1), X2) → U62¹(X1, X2)
ACTIVE(U61(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U11(X)) → ACTIVE(X)
PROPER(U61(X1, X2, X3)) → PROPER(X1)
TOP(ok(X)) → TOP(active(X))
PROPER(isNat(X)) → ISNAT(proper(X))
ACTIVE(U21(X)) → ACTIVE(X)
ISNAT(ok(X)) → ISNAT(X)
ACTIVE(U61(X1, X2, X3)) → U61¹(active(X1), X2, X3)
CONS(ok(X1), ok(X2)) → CONS(X1, X2)
ACTIVE(length(cons(N, L))) → ISNATLIST(L)
PROPER(U41(X1, X2)) → PROPER(X1)
PROPER(U52(X)) → U52¹(proper(X))
U52¹(ok(X)) → U52¹(X)
U11¹(ok(X)) → U11¹(X)
U42¹(mark(X)) → U42¹(X)
ACTIVE(U41(tt, V2)) → ISNATILIST(V2)
ACTIVE(U21(X)) → U21¹(active(X))
PROPER(U51(X1, X2)) → U51¹(proper(X1), proper(X2))
PROPER(U62(X1, X2)) → U62¹(proper(X1), proper(X2))
TOP(mark(X)) → TOP(proper(X))
ACTIVE(s(X)) → S(active(X))
PROPER(U31(X)) → U31¹(proper(X))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U21(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ DependencyPairsProof

                        ↳ QDP

                          ↳ DependencyGraphProof

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

ACTIVE(U51(X1, X2)) → ACTIVE(X1)
ACTIVE(U41(tt, V2)) → U42¹(isNatIList(V2))
PROPER(U51(X1, X2)) → PROPER(X2)
ACTIVE(U51(tt, V2)) → U52¹(isNatList(V2))
ACTIVE(U31(X)) → U31¹(active(X))
PROPER(U42(X)) → U42¹(proper(X))
ACTIVE(cons(X1, X2)) → CONS(active(X1), X2)
PROPER(U62(X1, X2)) → PROPER(X2)
ACTIVE(U61(tt, L, N)) → U62¹(isNat(N), L)
PROPER(isNatList(X)) → PROPER(X)
PROPER(length(X)) → LENGTH(proper(X))
ACTIVE(U52(X)) → U52¹(active(X))
U41¹(ok(X1), ok(X2)) → U41¹(X1, X2)
ACTIVE(U62(tt, L)) → S(length(L))
ACTIVE(isNatList(cons(V1, V2))) → U51¹(isNat(V1), V2)
PROPER(length(X)) → PROPER(X)
U62¹(ok(X1), ok(X2)) → U62¹(X1, X2)
PROPER(U61(X1, X2, X3)) → PROPER(X3)
ACTIVE(U41(X1, X2)) → U41¹(active(X1), X2)
PROPER(isNatList(X)) → ISNATLIST(proper(X))
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(U31(X)) → ACTIVE(X)
S(ok(X)) → S(X)
CONS(mark(X1), X2) → CONS(X1, X2)
TOP(mark(X)) → PROPER(X)
ACTIVE(U42(X)) → ACTIVE(X)
ACTIVE(isNatIList(cons(V1, V2))) → U41¹(isNat(V1), V2)
PROPER(U11(X)) → U11¹(proper(X))
TOP(ok(X)) → ACTIVE(X)
ACTIVE(length(X)) → ACTIVE(X)
ACTIVE(isNat(s(V1))) → U21¹(isNat(V1))
PROPER(cons(X1, X2)) → PROPER(X1)
ACTIVE(U41(X1, X2)) → ACTIVE(X1)
ACTIVE(U51(tt, V2)) → ISNATLIST(V2)
PROPER(U21(X)) → U21¹(proper(X))
LENGTH(mark(X)) → LENGTH(X)
ACTIVE(U62(tt, L)) → LENGTH(L)
U51¹(mark(X1), X2) → U51¹(X1, X2)
S(mark(X)) → S(X)
PROPER(cons(X1, X2)) → CONS(proper(X1), proper(X2))
PROPER(U41(X1, X2)) → PROPER(X2)
ACTIVE(length(X)) → LENGTH(active(X))
ISNATLIST(ok(X)) → ISNATLIST(X)
U11¹(mark(X)) → U11¹(X)
ACTIVE(U51(X1, X2)) → U51¹(active(X1), X2)
PROPER(U61(X1, X2, X3)) → PROPER(X2)
ACTIVE(isNat(s(V1))) → ISNAT(V1)
U61¹(mark(X1), X2, X3) → U61¹(X1, X2, X3)
PROPER(U51(X1, X2)) → PROPER(X1)
PROPER(U42(X)) → PROPER(X)
PROPER(U21(X)) → PROPER(X)
U51¹(ok(X1), ok(X2)) → U51¹(X1, X2)
U31¹(mark(X)) → U31¹(X)
PROPER(U62(X1, X2)) → PROPER(X1)
ACTIVE(U62(X1, X2)) → ACTIVE(X1)
ACTIVE(U11(X)) → U11¹(active(X))
PROPER(U61(X1, X2, X3)) → U61¹(proper(X1), proper(X2), proper(X3))
U31¹(ok(X)) → U31¹(X)
U41¹(mark(X1), X2) → U41¹(X1, X2)
U61¹(ok(X1), ok(X2), ok(X3)) → U61¹(X1, X2, X3)
U42¹(ok(X)) → U42¹(X)
U21¹(ok(X)) → U21¹(X)
ACTIVE(zeros) → CONS(0, zeros)
PROPER(cons(X1, X2)) → PROPER(X2)
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(isNatIList(cons(V1, V2))) → ISNAT(V1)
ISNATILIST(ok(X)) → ISNATILIST(X)
ACTIVE(U61(tt, L, N)) → ISNAT(N)
PROPER(U11(X)) → PROPER(X)
ACTIVE(U42(X)) → U42¹(active(X))
U21¹(mark(X)) → U21¹(X)
PROPER(s(X)) → S(proper(X))
ACTIVE(isNatList(cons(V1, V2))) → ISNAT(V1)
PROPER(U31(X)) → PROPER(X)
PROPER(isNatIList(X)) → ISNATILIST(proper(X))
PROPER(U52(X)) → PROPER(X)
ACTIVE(U62(X1, X2)) → U62¹(active(X1), X2)
ACTIVE(U52(X)) → ACTIVE(X)
U52¹(mark(X)) → U52¹(X)
ACTIVE(length(cons(N, L))) → U61¹(isNatList(L), L, N)
PROPER(isNat(X)) → PROPER(X)
LENGTH(ok(X)) → LENGTH(X)
PROPER(isNatIList(X)) → PROPER(X)
PROPER(U41(X1, X2)) → U41¹(proper(X1), proper(X2))
PROPER(s(X)) → PROPER(X)
U62¹(mark(X1), X2) → U62¹(X1, X2)
ACTIVE(U61(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U11(X)) → ACTIVE(X)
PROPER(U61(X1, X2, X3)) → PROPER(X1)
TOP(ok(X)) → TOP(active(X))
PROPER(isNat(X)) → ISNAT(proper(X))
ACTIVE(U21(X)) → ACTIVE(X)
ISNAT(ok(X)) → ISNAT(X)
ACTIVE(U61(X1, X2, X3)) → U61¹(active(X1), X2, X3)
CONS(ok(X1), ok(X2)) → CONS(X1, X2)
ACTIVE(length(cons(N, L))) → ISNATLIST(L)
PROPER(U41(X1, X2)) → PROPER(X1)
PROPER(U52(X)) → U52¹(proper(X))
U52¹(ok(X)) → U52¹(X)
U11¹(ok(X)) → U11¹(X)
U42¹(mark(X)) → U42¹(X)
ACTIVE(U41(tt, V2)) → ISNATILIST(V2)
ACTIVE(U21(X)) → U21¹(active(X))
PROPER(U51(X1, X2)) → U51¹(proper(X1), proper(X2))
PROPER(U62(X1, X2)) → U62¹(proper(X1), proper(X2))
TOP(mark(X)) → TOP(proper(X))
ACTIVE(s(X)) → S(active(X))
PROPER(U31(X)) → U31¹(proper(X))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U21(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 18 SCCs with 46 less nodes.

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ DependencyPairsProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                                ↳ UsableRulesProof

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

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

ISNAT(ok(X)) → ISNAT(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U21(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ DependencyPairsProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ QDPSizeChangeProof

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

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

ISNAT(ok(X)) → ISNAT(X)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

ISNAT(ok(X)) → ISNAT(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ DependencyPairsProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                              ↳ QDP

                                ↳ UsableRulesProof

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

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

ISNATLIST(ok(X)) → ISNATLIST(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U21(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ DependencyPairsProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ QDPSizeChangeProof

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

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

ISNATLIST(ok(X)) → ISNATLIST(X)

From the DPs we obtained the following set of size-change graphs:

ISNATLIST(ok(X)) → ISNATLIST(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ DependencyPairsProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                                ↳ UsableRulesProof

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

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

ISNATILIST(ok(X)) → ISNATILIST(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U21(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ DependencyPairsProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ QDPSizeChangeProof

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

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

ISNATILIST(ok(X)) → ISNATILIST(X)

From the DPs we obtained the following set of size-change graphs:

ISNATILIST(ok(X)) → ISNATILIST(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ DependencyPairsProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                                ↳ UsableRulesProof

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

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

LENGTH(ok(X)) → LENGTH(X)
LENGTH(mark(X)) → LENGTH(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U21(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ DependencyPairsProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ QDPSizeChangeProof

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

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

LENGTH(ok(X)) → LENGTH(X)
LENGTH(mark(X)) → LENGTH(X)

From the DPs we obtained the following set of size-change graphs:

LENGTH(ok(X)) → LENGTH(X)
The graph contains the following edges 1 > 1
LENGTH(mark(X)) → LENGTH(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ DependencyPairsProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                                ↳ UsableRulesProof

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

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

S(ok(X)) → S(X)
S(mark(X)) → S(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U21(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ DependencyPairsProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ QDPSizeChangeProof

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

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

S(ok(X)) → S(X)
S(mark(X)) → S(X)

From the DPs we obtained the following set of size-change graphs:

S(ok(X)) → S(X)
The graph contains the following edges 1 > 1
S(mark(X)) → S(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ DependencyPairsProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                                ↳ UsableRulesProof

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

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

U62¹(mark(X1), X2) → U62¹(X1, X2)
U62¹(ok(X1), ok(X2)) → U62¹(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U21(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ DependencyPairsProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ QDPSizeChangeProof

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

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

U62¹(mark(X1), X2) → U62¹(X1, X2)
U62¹(ok(X1), ok(X2)) → U62¹(X1, X2)

From the DPs we obtained the following set of size-change graphs:

U62¹(mark(X1), X2) → U62¹(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
U62¹(ok(X1), ok(X2)) → U62¹(X1, X2)
The graph contains the following edges 1 > 1, 2 > 2

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ DependencyPairsProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                                ↳ UsableRulesProof

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

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

U61¹(mark(X1), X2, X3) → U61¹(X1, X2, X3)
U61¹(ok(X1), ok(X2), ok(X3)) → U61¹(X1, X2, X3)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U21(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ DependencyPairsProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ QDPSizeChangeProof

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

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

U61¹(mark(X1), X2, X3) → U61¹(X1, X2, X3)
U61¹(ok(X1), ok(X2), ok(X3)) → U61¹(X1, X2, X3)

From the DPs we obtained the following set of size-change graphs:

U61¹(mark(X1), X2, X3) → U61¹(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
U61¹(ok(X1), ok(X2), ok(X3)) → U61¹(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 > 2, 3 > 3

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ DependencyPairsProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                                ↳ UsableRulesProof

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

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

U52¹(mark(X)) → U52¹(X)
U52¹(ok(X)) → U52¹(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U21(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ DependencyPairsProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ QDPSizeChangeProof

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

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

U52¹(mark(X)) → U52¹(X)
U52¹(ok(X)) → U52¹(X)

From the DPs we obtained the following set of size-change graphs:

U52¹(mark(X)) → U52¹(X)
The graph contains the following edges 1 > 1
U52¹(ok(X)) → U52¹(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ DependencyPairsProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                                ↳ UsableRulesProof

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

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

U51¹(ok(X1), ok(X2)) → U51¹(X1, X2)
U51¹(mark(X1), X2) → U51¹(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U21(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ DependencyPairsProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ QDPSizeChangeProof

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

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

U51¹(ok(X1), ok(X2)) → U51¹(X1, X2)
U51¹(mark(X1), X2) → U51¹(X1, X2)

From the DPs we obtained the following set of size-change graphs:

U51¹(ok(X1), ok(X2)) → U51¹(X1, X2)
The graph contains the following edges 1 > 1, 2 > 2
U51¹(mark(X1), X2) → U51¹(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ DependencyPairsProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                                ↳ UsableRulesProof

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

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

U42¹(mark(X)) → U42¹(X)
U42¹(ok(X)) → U42¹(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U21(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ DependencyPairsProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ QDPSizeChangeProof

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

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

U42¹(mark(X)) → U42¹(X)
U42¹(ok(X)) → U42¹(X)

From the DPs we obtained the following set of size-change graphs:

U42¹(mark(X)) → U42¹(X)
The graph contains the following edges 1 > 1
U42¹(ok(X)) → U42¹(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ DependencyPairsProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                                ↳ UsableRulesProof

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

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

U41¹(ok(X1), ok(X2)) → U41¹(X1, X2)
U41¹(mark(X1), X2) → U41¹(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U21(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ DependencyPairsProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ QDPSizeChangeProof

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

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

U41¹(ok(X1), ok(X2)) → U41¹(X1, X2)
U41¹(mark(X1), X2) → U41¹(X1, X2)

From the DPs we obtained the following set of size-change graphs:

U41¹(ok(X1), ok(X2)) → U41¹(X1, X2)
The graph contains the following edges 1 > 1, 2 > 2
U41¹(mark(X1), X2) → U41¹(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ DependencyPairsProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                                ↳ UsableRulesProof

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

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

U31¹(ok(X)) → U31¹(X)
U31¹(mark(X)) → U31¹(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U21(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ DependencyPairsProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ QDPSizeChangeProof

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

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

U31¹(ok(X)) → U31¹(X)
U31¹(mark(X)) → U31¹(X)

From the DPs we obtained the following set of size-change graphs:

U31¹(ok(X)) → U31¹(X)
The graph contains the following edges 1 > 1
U31¹(mark(X)) → U31¹(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ DependencyPairsProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                                ↳ UsableRulesProof

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

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

U21¹(ok(X)) → U21¹(X)
U21¹(mark(X)) → U21¹(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U21(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ DependencyPairsProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ QDPSizeChangeProof

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

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

U21¹(ok(X)) → U21¹(X)
U21¹(mark(X)) → U21¹(X)

From the DPs we obtained the following set of size-change graphs:

U21¹(ok(X)) → U21¹(X)
The graph contains the following edges 1 > 1
U21¹(mark(X)) → U21¹(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ DependencyPairsProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                                ↳ UsableRulesProof

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

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

U11¹(mark(X)) → U11¹(X)
U11¹(ok(X)) → U11¹(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U21(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ DependencyPairsProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ QDPSizeChangeProof

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

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

U11¹(mark(X)) → U11¹(X)
U11¹(ok(X)) → U11¹(X)

From the DPs we obtained the following set of size-change graphs:

U11¹(mark(X)) → U11¹(X)
The graph contains the following edges 1 > 1
U11¹(ok(X)) → U11¹(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ DependencyPairsProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                                ↳ UsableRulesProof

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

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

CONS(mark(X1), X2) → CONS(X1, X2)
CONS(ok(X1), ok(X2)) → CONS(X1, X2)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U21(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ DependencyPairsProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ QDPSizeChangeProof

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

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

CONS(mark(X1), X2) → CONS(X1, X2)
CONS(ok(X1), ok(X2)) → CONS(X1, X2)

From the DPs we obtained the following set of size-change graphs:

CONS(mark(X1), X2) → CONS(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
CONS(ok(X1), ok(X2)) → CONS(X1, X2)
The graph contains the following edges 1 > 1, 2 > 2

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ DependencyPairsProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                                ↳ UsableRulesProof

                              ↳ QDP

                              ↳ QDP

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

PROPER(isNat(X)) → PROPER(X)
PROPER(U11(X)) → PROPER(X)
PROPER(U51(X1, X2)) → PROPER(X2)
PROPER(U41(X1, X2)) → PROPER(X2)
PROPER(U41(X1, X2)) → PROPER(X1)
PROPER(length(X)) → PROPER(X)
PROPER(U61(X1, X2, X3)) → PROPER(X3)
PROPER(isNatIList(X)) → PROPER(X)
PROPER(U31(X)) → PROPER(X)
PROPER(U62(X1, X2)) → PROPER(X2)
PROPER(U61(X1, X2, X3)) → PROPER(X2)
PROPER(isNatList(X)) → PROPER(X)
PROPER(U51(X1, X2)) → PROPER(X1)
PROPER(s(X)) → PROPER(X)
PROPER(cons(X1, X2)) → PROPER(X1)
PROPER(U42(X)) → PROPER(X)
PROPER(U52(X)) → PROPER(X)
PROPER(U21(X)) → PROPER(X)
PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(U62(X1, X2)) → PROPER(X1)
PROPER(U61(X1, X2, X3)) → PROPER(X1)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U21(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ DependencyPairsProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ QDPSizeChangeProof

                              ↳ QDP

                              ↳ QDP

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

PROPER(isNat(X)) → PROPER(X)
PROPER(U11(X)) → PROPER(X)
PROPER(U51(X1, X2)) → PROPER(X2)
PROPER(U41(X1, X2)) → PROPER(X2)
PROPER(U41(X1, X2)) → PROPER(X1)
PROPER(length(X)) → PROPER(X)
PROPER(U61(X1, X2, X3)) → PROPER(X3)
PROPER(isNatIList(X)) → PROPER(X)
PROPER(U31(X)) → PROPER(X)
PROPER(U62(X1, X2)) → PROPER(X2)
PROPER(U61(X1, X2, X3)) → PROPER(X2)
PROPER(isNatList(X)) → PROPER(X)
PROPER(U51(X1, X2)) → PROPER(X1)
PROPER(U52(X)) → PROPER(X)
PROPER(U42(X)) → PROPER(X)
PROPER(cons(X1, X2)) → PROPER(X1)
PROPER(s(X)) → PROPER(X)
PROPER(U21(X)) → PROPER(X)
PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(U62(X1, X2)) → PROPER(X1)
PROPER(U61(X1, X2, X3)) → PROPER(X1)

From the DPs we obtained the following set of size-change graphs:

PROPER(isNat(X)) → PROPER(X)
The graph contains the following edges 1 > 1
PROPER(U11(X)) → PROPER(X)
The graph contains the following edges 1 > 1
PROPER(U51(X1, X2)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(U41(X1, X2)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(U41(X1, X2)) → PROPER(X1)
The graph contains the following edges 1 > 1
PROPER(length(X)) → PROPER(X)
The graph contains the following edges 1 > 1
PROPER(U61(X1, X2, X3)) → PROPER(X3)
The graph contains the following edges 1 > 1
PROPER(isNatIList(X)) → PROPER(X)
The graph contains the following edges 1 > 1
PROPER(U31(X)) → PROPER(X)
The graph contains the following edges 1 > 1
PROPER(U62(X1, X2)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(U61(X1, X2, X3)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(isNatList(X)) → PROPER(X)
The graph contains the following edges 1 > 1
PROPER(U51(X1, X2)) → PROPER(X1)
The graph contains the following edges 1 > 1
PROPER(s(X)) → PROPER(X)
The graph contains the following edges 1 > 1
PROPER(cons(X1, X2)) → PROPER(X1)
The graph contains the following edges 1 > 1
PROPER(U42(X)) → PROPER(X)
The graph contains the following edges 1 > 1
PROPER(U52(X)) → PROPER(X)
The graph contains the following edges 1 > 1
PROPER(U21(X)) → PROPER(X)
The graph contains the following edges 1 > 1
PROPER(cons(X1, X2)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(U62(X1, X2)) → PROPER(X1)
The graph contains the following edges 1 > 1
PROPER(U61(X1, X2, X3)) → PROPER(X1)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ DependencyPairsProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                                ↳ UsableRulesProof

                              ↳ QDP

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

ACTIVE(U51(X1, X2)) → ACTIVE(X1)
ACTIVE(U21(X)) → ACTIVE(X)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(U41(X1, X2)) → ACTIVE(X1)
ACTIVE(U42(X)) → ACTIVE(X)
ACTIVE(U61(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(U11(X)) → ACTIVE(X)
ACTIVE(U31(X)) → ACTIVE(X)
ACTIVE(length(X)) → ACTIVE(X)
ACTIVE(U62(X1, X2)) → ACTIVE(X1)
ACTIVE(U52(X)) → ACTIVE(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U21(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ DependencyPairsProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ QDPSizeChangeProof

                              ↳ QDP

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

ACTIVE(U21(X)) → ACTIVE(X)
ACTIVE(U51(X1, X2)) → ACTIVE(X1)
ACTIVE(U41(X1, X2)) → ACTIVE(X1)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(U42(X)) → ACTIVE(X)
ACTIVE(U61(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(U11(X)) → ACTIVE(X)
ACTIVE(U31(X)) → ACTIVE(X)
ACTIVE(U52(X)) → ACTIVE(X)
ACTIVE(U62(X1, X2)) → ACTIVE(X1)
ACTIVE(length(X)) → ACTIVE(X)

From the DPs we obtained the following set of size-change graphs:

ACTIVE(U51(X1, X2)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
ACTIVE(U21(X)) → ACTIVE(X)
The graph contains the following edges 1 > 1
ACTIVE(s(X)) → ACTIVE(X)
The graph contains the following edges 1 > 1
ACTIVE(U41(X1, X2)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
ACTIVE(U42(X)) → ACTIVE(X)
The graph contains the following edges 1 > 1
ACTIVE(U61(X1, X2, X3)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
ACTIVE(U11(X)) → ACTIVE(X)
The graph contains the following edges 1 > 1
ACTIVE(U31(X)) → ACTIVE(X)
The graph contains the following edges 1 > 1
ACTIVE(length(X)) → ACTIVE(X)
The graph contains the following edges 1 > 1
ACTIVE(U62(X1, X2)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
ACTIVE(U52(X)) → ACTIVE(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ DependencyPairsProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                                ↳ UsableRulesReductionPairsProof

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

TOP(mark(X)) → TOP(proper(X))
TOP(ok(X)) → TOP(active(X))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U21(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2) → mark(U62(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(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(TOP(x₁)) = x₁
POL(U11(x₁)) = x₁
POL(U21(x₁)) = x₁
POL(U31(x₁)) = 2·x₁
POL(U41(x₁, x₂)) = x₁ + 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)) = x₁ + 2·x₂
POL(active(x₁)) = x₁
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(nil) = 0
POL(ok(x₁)) = 2·x₁
POL(proper(x₁)) = x₁
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ DependencyPairsProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                                ↳ UsableRulesReductionPairsProof

                                  ↳ QDP

                                    ↳ Narrowing

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

TOP(mark(X)) → TOP(proper(X))
TOP(ok(X)) → TOP(active(X))

The TRS R consists of the following rules:

proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
isNat(ok(X)) → ok(isNat(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
isNatList(ok(X)) → ok(isNatList(X))
U52(mark(X)) → mark(U52(X))
U52(ok(X)) → ok(U52(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
U42(mark(X)) → mark(U42(X))
U42(ok(X)) → ok(U42(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U31(mark(X)) → mark(U31(X))
U31(ok(X)) → ok(U31(X))
U21(mark(X)) → mark(U21(X))
U21(ok(X)) → ok(U21(X))
U11(mark(X)) → mark(U11(X))
U11(ok(X)) → ok(U11(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
active(zeros) → mark(cons(0, zeros))
active(U21(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))

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

TOP(mark(U11(x0))) → TOP(U11(proper(x0)))
TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(mark(zeros)) → TOP(ok(zeros))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U31(x0))) → TOP(U31(proper(x0)))
TOP(mark(U52(x0))) → TOP(U52(proper(x0)))
TOP(mark(U41(x0, x1))) → TOP(U41(proper(x0), proper(x1)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(mark(0)) → TOP(ok(0))
TOP(mark(tt)) → TOP(ok(tt))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(mark(U42(x0))) → TOP(U42(proper(x0)))
TOP(mark(nil)) → TOP(ok(nil))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))
TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ DependencyPairsProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                                ↳ UsableRulesReductionPairsProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

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

TOP(mark(U11(x0))) → TOP(U11(proper(x0)))
TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(mark(zeros)) → TOP(ok(zeros))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U31(x0))) → TOP(U31(proper(x0)))
TOP(mark(U52(x0))) → TOP(U52(proper(x0)))
TOP(mark(U41(x0, x1))) → TOP(U41(proper(x0), proper(x1)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(mark(0)) → TOP(ok(0))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(mark(tt)) → TOP(ok(tt))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(mark(U42(x0))) → TOP(U42(proper(x0)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(nil)) → TOP(ok(nil))
TOP(ok(X)) → TOP(active(X))
TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))

The TRS R consists of the following rules:

proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
isNat(ok(X)) → ok(isNat(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
isNatList(ok(X)) → ok(isNatList(X))
U52(mark(X)) → mark(U52(X))
U52(ok(X)) → ok(U52(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
U42(mark(X)) → mark(U42(X))
U42(ok(X)) → ok(U42(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U31(mark(X)) → mark(U31(X))
U31(ok(X)) → ok(U31(X))
U21(mark(X)) → mark(U21(X))
U21(ok(X)) → ok(U21(X))
U11(mark(X)) → mark(U11(X))
U11(ok(X)) → ok(U11(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
active(zeros) → mark(cons(0, zeros))
active(U21(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 3 less nodes.

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ DependencyPairsProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                                ↳ UsableRulesReductionPairsProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Narrowing

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

TOP(mark(U11(x0))) → TOP(U11(proper(x0)))
TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U31(x0))) → TOP(U31(proper(x0)))
TOP(mark(U52(x0))) → TOP(U52(proper(x0)))
TOP(mark(U41(x0, x1))) → TOP(U41(proper(x0), proper(x1)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(mark(tt)) → TOP(ok(tt))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(mark(U42(x0))) → TOP(U42(proper(x0)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(X)) → TOP(active(X))
TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))

The TRS R consists of the following rules:

proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
isNat(ok(X)) → ok(isNat(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
isNatList(ok(X)) → ok(isNatList(X))
U52(mark(X)) → mark(U52(X))
U52(ok(X)) → ok(U52(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
U42(mark(X)) → mark(U42(X))
U42(ok(X)) → ok(U42(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U31(mark(X)) → mark(U31(X))
U31(ok(X)) → ok(U31(X))
U21(mark(X)) → mark(U21(X))
U21(ok(X)) → ok(U21(X))
U11(mark(X)) → mark(U11(X))
U11(ok(X)) → ok(U11(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
active(zeros) → mark(cons(0, zeros))
active(U21(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))

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

TOP(ok(isNat(s(x0)))) → TOP(mark(U21(isNat(x0))))
TOP(ok(isNatIList(cons(x0, x1)))) → TOP(mark(U41(isNat(x0), x1)))
TOP(ok(U52(x0))) → TOP(U52(active(x0)))
TOP(ok(U62(tt, x0))) → TOP(mark(s(length(x0))))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(isNatList(cons(x0, x1)))) → TOP(mark(U51(isNat(x0), x1)))
TOP(ok(U52(tt))) → TOP(mark(tt))
TOP(ok(zeros)) → TOP(mark(cons(0, zeros)))
TOP(ok(U41(tt, x0))) → TOP(mark(U42(isNatIList(x0))))
TOP(ok(U42(tt))) → TOP(mark(tt))
TOP(ok(U51(tt, x0))) → TOP(mark(U52(isNatList(x0))))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x1), x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(ok(U11(x0))) → TOP(U11(active(x0)))
TOP(ok(U42(x0))) → TOP(U42(active(x0)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(ok(isNat(0))) → TOP(mark(tt))
TOP(ok(U41(x0, x1))) → TOP(U41(active(x0), x1))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(U31(x0))) → TOP(U31(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U61(isNatList(x1), x1, x0)))
TOP(ok(U21(tt))) → TOP(mark(tt))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ DependencyPairsProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                                ↳ UsableRulesReductionPairsProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

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

TOP(mark(U11(x0))) → TOP(U11(proper(x0)))
TOP(ok(isNat(s(x0)))) → TOP(mark(U21(isNat(x0))))
TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(ok(isNatIList(cons(x0, x1)))) → TOP(mark(U41(isNat(x0), x1)))
TOP(ok(U52(x0))) → TOP(U52(active(x0)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(ok(U62(tt, x0))) → TOP(mark(s(length(x0))))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(isNatList(cons(x0, x1)))) → TOP(mark(U51(isNat(x0), x1)))
TOP(mark(tt)) → TOP(ok(tt))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(U52(tt))) → TOP(mark(tt))
TOP(ok(zeros)) → TOP(mark(cons(0, zeros)))
TOP(ok(U41(tt, x0))) → TOP(mark(U42(isNatIList(x0))))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(U51(tt, x0))) → TOP(mark(U52(isNatList(x0))))
TOP(ok(U42(tt))) → TOP(mark(tt))
TOP(mark(U42(x0))) → TOP(U42(proper(x0)))
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x1), x0)))
TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(ok(U42(x0))) → TOP(U42(active(x0)))
TOP(ok(U11(x0))) → TOP(U11(active(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U31(x0))) → TOP(U31(proper(x0)))
TOP(mark(U52(x0))) → TOP(U52(proper(x0)))
TOP(ok(isNat(0))) → TOP(mark(tt))
TOP(mark(U41(x0, x1))) → TOP(U41(proper(x0), proper(x1)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U41(x0, x1))) → TOP(U41(active(x0), x1))
TOP(ok(U31(x0))) → TOP(U31(active(x0)))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U21(tt))) → TOP(mark(tt))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U61(isNatList(x1), x1, x0)))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))

The TRS R consists of the following rules:

proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
isNat(ok(X)) → ok(isNat(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
isNatList(ok(X)) → ok(isNatList(X))
U52(mark(X)) → mark(U52(X))
U52(ok(X)) → ok(U52(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
U42(mark(X)) → mark(U42(X))
U42(ok(X)) → ok(U42(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U31(mark(X)) → mark(U31(X))
U31(ok(X)) → ok(U31(X))
U21(mark(X)) → mark(U21(X))
U21(ok(X)) → ok(U21(X))
U11(mark(X)) → mark(U11(X))
U11(ok(X)) → ok(U11(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
active(zeros) → mark(cons(0, zeros))
active(U21(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 5 less nodes.

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ DependencyPairsProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                                ↳ UsableRulesReductionPairsProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

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

TOP(mark(U11(x0))) → TOP(U11(proper(x0)))
TOP(ok(isNat(s(x0)))) → TOP(mark(U21(isNat(x0))))
TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(ok(isNatIList(cons(x0, x1)))) → TOP(mark(U41(isNat(x0), x1)))
TOP(ok(U52(x0))) → TOP(U52(active(x0)))
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(ok(U62(tt, x0))) → TOP(mark(s(length(x0))))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(isNatList(cons(x0, x1)))) → TOP(mark(U51(isNat(x0), x1)))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(zeros)) → TOP(mark(cons(0, zeros)))
TOP(ok(U41(tt, x0))) → TOP(mark(U42(isNatIList(x0))))
TOP(mark(U42(x0))) → TOP(U42(proper(x0)))
TOP(ok(U51(tt, x0))) → TOP(mark(U52(isNatList(x0))))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x1), x0)))
TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(ok(U42(x0))) → TOP(U42(active(x0)))
TOP(ok(U11(x0))) → TOP(U11(active(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U31(x0))) → TOP(U31(proper(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U52(x0))) → TOP(U52(proper(x0)))
TOP(mark(U41(x0, x1))) → TOP(U41(proper(x0), proper(x1)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U41(x0, x1))) → TOP(U41(active(x0), x1))
TOP(ok(U31(x0))) → TOP(U31(active(x0)))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U61(isNatList(x1), x1, x0)))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))

The TRS R consists of the following rules:

proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
isNat(ok(X)) → ok(isNat(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
isNatList(ok(X)) → ok(isNatList(X))
U52(mark(X)) → mark(U52(X))
U52(ok(X)) → ok(U52(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
U42(mark(X)) → mark(U42(X))
U42(ok(X)) → ok(U42(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U31(mark(X)) → mark(U31(X))
U31(ok(X)) → ok(U31(X))
U21(mark(X)) → mark(U21(X))
U21(ok(X)) → ok(U21(X))
U11(mark(X)) → mark(U11(X))
U11(ok(X)) → ok(U11(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
active(zeros) → mark(cons(0, zeros))
active(U21(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))

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

The following pairs can be oriented strictly and are deleted.

TOP(ok(zeros)) → TOP(mark(cons(0, zeros)))

The remaining pairs can at least be oriented weakly.

TOP(mark(U11(x0))) → TOP(U11(proper(x0)))
TOP(ok(isNat(s(x0)))) → TOP(mark(U21(isNat(x0))))
TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(ok(isNatIList(cons(x0, x1)))) → TOP(mark(U41(isNat(x0), x1)))
TOP(ok(U52(x0))) → TOP(U52(active(x0)))
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(ok(U62(tt, x0))) → TOP(mark(s(length(x0))))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(isNatList(cons(x0, x1)))) → TOP(mark(U51(isNat(x0), x1)))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(U41(tt, x0))) → TOP(mark(U42(isNatIList(x0))))
TOP(mark(U42(x0))) → TOP(U42(proper(x0)))
TOP(ok(U51(tt, x0))) → TOP(mark(U52(isNatList(x0))))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x1), x0)))
TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(ok(U42(x0))) → TOP(U42(active(x0)))
TOP(ok(U11(x0))) → TOP(U11(active(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U31(x0))) → TOP(U31(proper(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U52(x0))) → TOP(U52(proper(x0)))
TOP(mark(U41(x0, x1))) → TOP(U41(proper(x0), proper(x1)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U41(x0, x1))) → TOP(U41(active(x0), x1))
TOP(ok(U31(x0))) → TOP(U31(active(x0)))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U61(isNatList(x1), x1, x0)))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(TOP(x₁)) = x₁
POL(U11(x₁)) = 0
POL(U21(x₁)) = 0
POL(U31(x₁)) = 0
POL(U41(x₁, x₂)) = 0
POL(U42(x₁)) = 0
POL(U51(x₁, x₂)) = 0
POL(U52(x₁)) = 0
POL(U61(x₁, x₂, x₃)) = 0
POL(U62(x₁, x₂)) = 0
POL(active(x₁)) = 0
POL(cons(x₁, x₂)) = 0
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 0
POL(mark(x₁)) = 0
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = 0
POL(s(x₁)) = 0
POL(tt) = 0
POL(zeros) = 1

The following usable rules [17] were oriented:

U11(ok(X)) → ok(U11(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U21(ok(X)) → ok(U21(X))
U11(mark(X)) → mark(U11(X))
U31(ok(X)) → ok(U31(X))
U21(mark(X)) → mark(U21(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U31(mark(X)) → mark(U31(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
s(ok(X)) → ok(s(X))
s(mark(X)) → mark(s(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U42(mark(X)) → mark(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U52(ok(X)) → ok(U52(X))
U52(mark(X)) → mark(U52(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ DependencyPairsProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                                ↳ UsableRulesReductionPairsProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

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

TOP(ok(isNat(s(x0)))) → TOP(mark(U21(isNat(x0))))
TOP(mark(U11(x0))) → TOP(U11(proper(x0)))
TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(ok(isNatIList(cons(x0, x1)))) → TOP(mark(U41(isNat(x0), x1)))
TOP(ok(U52(x0))) → TOP(U52(active(x0)))
TOP(ok(U62(tt, x0))) → TOP(mark(s(length(x0))))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(isNatList(cons(x0, x1)))) → TOP(mark(U51(isNat(x0), x1)))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(U41(tt, x0))) → TOP(mark(U42(isNatIList(x0))))
TOP(mark(U42(x0))) → TOP(U42(proper(x0)))
TOP(ok(U51(tt, x0))) → TOP(mark(U52(isNatList(x0))))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x1), x0)))
TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(ok(U42(x0))) → TOP(U42(active(x0)))
TOP(ok(U11(x0))) → TOP(U11(active(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U31(x0))) → TOP(U31(proper(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U52(x0))) → TOP(U52(proper(x0)))
TOP(mark(U41(x0, x1))) → TOP(U41(proper(x0), proper(x1)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U41(x0, x1))) → TOP(U41(active(x0), x1))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(U31(x0))) → TOP(U31(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U61(isNatList(x1), x1, x0)))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))

The TRS R consists of the following rules:

proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
isNat(ok(X)) → ok(isNat(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
isNatList(ok(X)) → ok(isNatList(X))
U52(mark(X)) → mark(U52(X))
U52(ok(X)) → ok(U52(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
U42(mark(X)) → mark(U42(X))
U42(ok(X)) → ok(U42(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U31(mark(X)) → mark(U31(X))
U31(ok(X)) → ok(U31(X))
U21(mark(X)) → mark(U21(X))
U21(ok(X)) → ok(U21(X))
U11(mark(X)) → mark(U11(X))
U11(ok(X)) → ok(U11(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
active(zeros) → mark(cons(0, zeros))
active(U21(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))

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

The following pairs can be oriented strictly and are deleted.

TOP(ok(isNatList(cons(x0, x1)))) → TOP(mark(U51(isNat(x0), x1)))
TOP(ok(U41(tt, x0))) → TOP(mark(U42(isNatIList(x0))))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U61(isNatList(x1), x1, x0)))

The remaining pairs can at least be oriented weakly.

TOP(ok(isNat(s(x0)))) → TOP(mark(U21(isNat(x0))))
TOP(mark(U11(x0))) → TOP(U11(proper(x0)))
TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(ok(isNatIList(cons(x0, x1)))) → TOP(mark(U41(isNat(x0), x1)))
TOP(ok(U52(x0))) → TOP(U52(active(x0)))
TOP(ok(U62(tt, x0))) → TOP(mark(s(length(x0))))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(mark(U42(x0))) → TOP(U42(proper(x0)))
TOP(ok(U51(tt, x0))) → TOP(mark(U52(isNatList(x0))))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x1), x0)))
TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(ok(U42(x0))) → TOP(U42(active(x0)))
TOP(ok(U11(x0))) → TOP(U11(active(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U31(x0))) → TOP(U31(proper(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U52(x0))) → TOP(U52(proper(x0)))
TOP(mark(U41(x0, x1))) → TOP(U41(proper(x0), proper(x1)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U41(x0, x1))) → TOP(U41(active(x0), x1))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(U31(x0))) → TOP(U31(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(TOP(x₁)) = x₁
POL(U11(x₁)) = 0
POL(U21(x₁)) = 0
POL(U31(x₁)) = 0
POL(U41(x₁, x₂)) = 1
POL(U42(x₁)) = 0
POL(U51(x₁, x₂)) = 0
POL(U52(x₁)) = 0
POL(U61(x₁, x₂, x₃)) = 0
POL(U62(x₁, x₂)) = 0
POL(active(x₁)) = 0
POL(cons(x₁, x₂)) = 1
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = x₁
POL(isNatList(x₁)) = 1 + x₁
POL(length(x₁)) = 1
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = x₁
POL(s(x₁)) = 0
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

U11(ok(X)) → ok(U11(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U21(ok(X)) → ok(U21(X))
U11(mark(X)) → mark(U11(X))
U31(ok(X)) → ok(U31(X))
U21(mark(X)) → mark(U21(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U31(mark(X)) → mark(U31(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
s(ok(X)) → ok(s(X))
s(mark(X)) → mark(s(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U42(mark(X)) → mark(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U52(ok(X)) → ok(U52(X))
U52(mark(X)) → mark(U52(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(zeros) → ok(zeros)

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ DependencyPairsProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                                ↳ UsableRulesReductionPairsProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

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

TOP(mark(U11(x0))) → TOP(U11(proper(x0)))
TOP(ok(isNat(s(x0)))) → TOP(mark(U21(isNat(x0))))
TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(ok(isNatIList(cons(x0, x1)))) → TOP(mark(U41(isNat(x0), x1)))
TOP(ok(U52(x0))) → TOP(U52(active(x0)))
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(ok(U62(tt, x0))) → TOP(mark(s(length(x0))))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(U51(tt, x0))) → TOP(mark(U52(isNatList(x0))))
TOP(mark(U42(x0))) → TOP(U42(proper(x0)))
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x1), x0)))
TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(ok(U42(x0))) → TOP(U42(active(x0)))
TOP(ok(U11(x0))) → TOP(U11(active(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U31(x0))) → TOP(U31(proper(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U52(x0))) → TOP(U52(proper(x0)))
TOP(mark(U41(x0, x1))) → TOP(U41(proper(x0), proper(x1)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U41(x0, x1))) → TOP(U41(active(x0), x1))
TOP(ok(U31(x0))) → TOP(U31(active(x0)))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))

The TRS R consists of the following rules:

proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
isNat(ok(X)) → ok(isNat(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
isNatList(ok(X)) → ok(isNatList(X))
U52(mark(X)) → mark(U52(X))
U52(ok(X)) → ok(U52(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
U42(mark(X)) → mark(U42(X))
U42(ok(X)) → ok(U42(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U31(mark(X)) → mark(U31(X))
U31(ok(X)) → ok(U31(X))
U21(mark(X)) → mark(U21(X))
U21(ok(X)) → ok(U21(X))
U11(mark(X)) → mark(U11(X))
U11(ok(X)) → ok(U11(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
active(zeros) → mark(cons(0, zeros))
active(U21(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))

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

The following pairs can be oriented strictly and are deleted.

TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))

The remaining pairs can at least be oriented weakly.

TOP(mark(U11(x0))) → TOP(U11(proper(x0)))
TOP(ok(isNat(s(x0)))) → TOP(mark(U21(isNat(x0))))
TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(ok(isNatIList(cons(x0, x1)))) → TOP(mark(U41(isNat(x0), x1)))
TOP(ok(U52(x0))) → TOP(U52(active(x0)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(ok(U62(tt, x0))) → TOP(mark(s(length(x0))))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(U51(tt, x0))) → TOP(mark(U52(isNatList(x0))))
TOP(mark(U42(x0))) → TOP(U42(proper(x0)))
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x1), x0)))
TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(ok(U42(x0))) → TOP(U42(active(x0)))
TOP(ok(U11(x0))) → TOP(U11(active(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U31(x0))) → TOP(U31(proper(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U52(x0))) → TOP(U52(proper(x0)))
TOP(mark(U41(x0, x1))) → TOP(U41(proper(x0), proper(x1)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U41(x0, x1))) → TOP(U41(active(x0), x1))
TOP(ok(U31(x0))) → TOP(U31(active(x0)))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(TOP(x₁)) = x₁
POL(U11(x₁)) = 1
POL(U21(x₁)) = 1
POL(U31(x₁)) = 1
POL(U41(x₁, x₂)) = 1
POL(U42(x₁)) = 1
POL(U51(x₁, x₂)) = 1
POL(U52(x₁)) = 1
POL(U61(x₁, x₂, x₃)) = 1
POL(U62(x₁, x₂)) = 1
POL(active(x₁)) = 0
POL(cons(x₁, x₂)) = 1
POL(isNat(x₁)) = 1
POL(isNatIList(x₁)) = 1
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 1
POL(mark(x₁)) = 1
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = 0
POL(s(x₁)) = 1
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

U11(ok(X)) → ok(U11(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U21(ok(X)) → ok(U21(X))
U11(mark(X)) → mark(U11(X))
U31(ok(X)) → ok(U31(X))
U21(mark(X)) → mark(U21(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U31(mark(X)) → mark(U31(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
s(ok(X)) → ok(s(X))
s(mark(X)) → mark(s(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U42(mark(X)) → mark(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U52(ok(X)) → ok(U52(X))
U52(mark(X)) → mark(U52(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ DependencyPairsProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                                ↳ UsableRulesReductionPairsProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

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

TOP(ok(isNat(s(x0)))) → TOP(mark(U21(isNat(x0))))
TOP(mark(U11(x0))) → TOP(U11(proper(x0)))
TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(ok(isNatIList(cons(x0, x1)))) → TOP(mark(U41(isNat(x0), x1)))
TOP(ok(U52(x0))) → TOP(U52(active(x0)))
TOP(ok(U62(tt, x0))) → TOP(mark(s(length(x0))))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(mark(U42(x0))) → TOP(U42(proper(x0)))
TOP(ok(U51(tt, x0))) → TOP(mark(U52(isNatList(x0))))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x1), x0)))
TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(ok(U42(x0))) → TOP(U42(active(x0)))
TOP(ok(U11(x0))) → TOP(U11(active(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U31(x0))) → TOP(U31(proper(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U52(x0))) → TOP(U52(proper(x0)))
TOP(mark(U41(x0, x1))) → TOP(U41(proper(x0), proper(x1)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U41(x0, x1))) → TOP(U41(active(x0), x1))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(U31(x0))) → TOP(U31(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))

The TRS R consists of the following rules:

proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
isNat(ok(X)) → ok(isNat(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
isNatList(ok(X)) → ok(isNatList(X))
U52(mark(X)) → mark(U52(X))
U52(ok(X)) → ok(U52(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
U42(mark(X)) → mark(U42(X))
U42(ok(X)) → ok(U42(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U31(mark(X)) → mark(U31(X))
U31(ok(X)) → ok(U31(X))
U21(mark(X)) → mark(U21(X))
U21(ok(X)) → ok(U21(X))
U11(mark(X)) → mark(U11(X))
U11(ok(X)) → ok(U11(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
active(zeros) → mark(cons(0, zeros))
active(U21(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))

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

The following pairs can be oriented strictly and are deleted.

TOP(ok(U61(tt, x0, x1))) → TOP(mark(U62(isNat(x1), x0)))

The remaining pairs can at least be oriented weakly.

TOP(ok(isNat(s(x0)))) → TOP(mark(U21(isNat(x0))))
TOP(mark(U11(x0))) → TOP(U11(proper(x0)))
TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(ok(isNatIList(cons(x0, x1)))) → TOP(mark(U41(isNat(x0), x1)))
TOP(ok(U52(x0))) → TOP(U52(active(x0)))
TOP(ok(U62(tt, x0))) → TOP(mark(s(length(x0))))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(mark(U42(x0))) → TOP(U42(proper(x0)))
TOP(ok(U51(tt, x0))) → TOP(mark(U52(isNatList(x0))))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(ok(U42(x0))) → TOP(U42(active(x0)))
TOP(ok(U11(x0))) → TOP(U11(active(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U31(x0))) → TOP(U31(proper(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U52(x0))) → TOP(U52(proper(x0)))
TOP(mark(U41(x0, x1))) → TOP(U41(proper(x0), proper(x1)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U41(x0, x1))) → TOP(U41(active(x0), x1))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(U31(x0))) → TOP(U31(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(TOP(x₁)) = x₁
POL(U11(x₁)) = 0
POL(U21(x₁)) = 0
POL(U31(x₁)) = 0
POL(U41(x₁, x₂)) = 0
POL(U42(x₁)) = 0
POL(U51(x₁, x₂)) = 0
POL(U52(x₁)) = 0
POL(U61(x₁, x₂, x₃)) = 1
POL(U62(x₁, x₂)) = 0
POL(active(x₁)) = 0
POL(cons(x₁, x₂)) = 0
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = 0
POL(s(x₁)) = 0
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

U11(ok(X)) → ok(U11(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U21(ok(X)) → ok(U21(X))
U11(mark(X)) → mark(U11(X))
U31(ok(X)) → ok(U31(X))
U21(mark(X)) → mark(U21(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U31(mark(X)) → mark(U31(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
s(ok(X)) → ok(s(X))
s(mark(X)) → mark(s(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U42(mark(X)) → mark(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U52(ok(X)) → ok(U52(X))
U52(mark(X)) → mark(U52(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ DependencyPairsProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                                ↳ UsableRulesReductionPairsProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

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

TOP(mark(U11(x0))) → TOP(U11(proper(x0)))
TOP(ok(isNat(s(x0)))) → TOP(mark(U21(isNat(x0))))
TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(ok(isNatIList(cons(x0, x1)))) → TOP(mark(U41(isNat(x0), x1)))
TOP(ok(U52(x0))) → TOP(U52(active(x0)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(ok(U62(tt, x0))) → TOP(mark(s(length(x0))))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(U51(tt, x0))) → TOP(mark(U52(isNatList(x0))))
TOP(mark(U42(x0))) → TOP(U42(proper(x0)))
TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(ok(U42(x0))) → TOP(U42(active(x0)))
TOP(ok(U11(x0))) → TOP(U11(active(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U31(x0))) → TOP(U31(proper(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U52(x0))) → TOP(U52(proper(x0)))
TOP(mark(U41(x0, x1))) → TOP(U41(proper(x0), proper(x1)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U41(x0, x1))) → TOP(U41(active(x0), x1))
TOP(ok(U31(x0))) → TOP(U31(active(x0)))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))

The TRS R consists of the following rules:

proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
isNat(ok(X)) → ok(isNat(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
isNatList(ok(X)) → ok(isNatList(X))
U52(mark(X)) → mark(U52(X))
U52(ok(X)) → ok(U52(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
U42(mark(X)) → mark(U42(X))
U42(ok(X)) → ok(U42(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U31(mark(X)) → mark(U31(X))
U31(ok(X)) → ok(U31(X))
U21(mark(X)) → mark(U21(X))
U21(ok(X)) → ok(U21(X))
U11(mark(X)) → mark(U11(X))
U11(ok(X)) → ok(U11(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
active(zeros) → mark(cons(0, zeros))
active(U21(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))

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

The following pairs can be oriented strictly and are deleted.

TOP(ok(U51(tt, x0))) → TOP(mark(U52(isNatList(x0))))

The remaining pairs can at least be oriented weakly.

TOP(mark(U11(x0))) → TOP(U11(proper(x0)))
TOP(ok(isNat(s(x0)))) → TOP(mark(U21(isNat(x0))))
TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(ok(isNatIList(cons(x0, x1)))) → TOP(mark(U41(isNat(x0), x1)))
TOP(ok(U52(x0))) → TOP(U52(active(x0)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(ok(U62(tt, x0))) → TOP(mark(s(length(x0))))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U42(x0))) → TOP(U42(proper(x0)))
TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(ok(U42(x0))) → TOP(U42(active(x0)))
TOP(ok(U11(x0))) → TOP(U11(active(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U31(x0))) → TOP(U31(proper(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U52(x0))) → TOP(U52(proper(x0)))
TOP(mark(U41(x0, x1))) → TOP(U41(proper(x0), proper(x1)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U41(x0, x1))) → TOP(U41(active(x0), x1))
TOP(ok(U31(x0))) → TOP(U31(active(x0)))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(TOP(x₁)) = x₁
POL(U11(x₁)) = 0
POL(U21(x₁)) = 0
POL(U31(x₁)) = 0
POL(U41(x₁, x₂)) = 0
POL(U42(x₁)) = 0
POL(U51(x₁, x₂)) = 1
POL(U52(x₁)) = 0
POL(U61(x₁, x₂, x₃)) = 0
POL(U62(x₁, x₂)) = 0
POL(active(x₁)) = 0
POL(cons(x₁, x₂)) = 0
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = 0
POL(s(x₁)) = 0
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

U11(ok(X)) → ok(U11(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U21(ok(X)) → ok(U21(X))
U11(mark(X)) → mark(U11(X))
U31(ok(X)) → ok(U31(X))
U21(mark(X)) → mark(U21(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U31(mark(X)) → mark(U31(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
s(ok(X)) → ok(s(X))
s(mark(X)) → mark(s(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U42(mark(X)) → mark(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U52(ok(X)) → ok(U52(X))
U52(mark(X)) → mark(U52(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ DependencyPairsProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                                ↳ UsableRulesReductionPairsProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

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

TOP(ok(isNat(s(x0)))) → TOP(mark(U21(isNat(x0))))
TOP(mark(U11(x0))) → TOP(U11(proper(x0)))
TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(ok(isNatIList(cons(x0, x1)))) → TOP(mark(U41(isNat(x0), x1)))
TOP(ok(U52(x0))) → TOP(U52(active(x0)))
TOP(ok(U62(tt, x0))) → TOP(mark(s(length(x0))))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(mark(U42(x0))) → TOP(U42(proper(x0)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(ok(U42(x0))) → TOP(U42(active(x0)))
TOP(ok(U11(x0))) → TOP(U11(active(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U31(x0))) → TOP(U31(proper(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U52(x0))) → TOP(U52(proper(x0)))
TOP(mark(U41(x0, x1))) → TOP(U41(proper(x0), proper(x1)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U41(x0, x1))) → TOP(U41(active(x0), x1))
TOP(ok(U31(x0))) → TOP(U31(active(x0)))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))

The TRS R consists of the following rules:

proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
isNat(ok(X)) → ok(isNat(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
isNatList(ok(X)) → ok(isNatList(X))
U52(mark(X)) → mark(U52(X))
U52(ok(X)) → ok(U52(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
U42(mark(X)) → mark(U42(X))
U42(ok(X)) → ok(U42(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U31(mark(X)) → mark(U31(X))
U31(ok(X)) → ok(U31(X))
U21(mark(X)) → mark(U21(X))
U21(ok(X)) → ok(U21(X))
U11(mark(X)) → mark(U11(X))
U11(ok(X)) → ok(U11(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
active(zeros) → mark(cons(0, zeros))
active(U21(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))

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

The following pairs can be oriented strictly and are deleted.

TOP(ok(isNat(s(x0)))) → TOP(mark(U21(isNat(x0))))

The remaining pairs can at least be oriented weakly.

TOP(mark(U11(x0))) → TOP(U11(proper(x0)))
TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(ok(isNatIList(cons(x0, x1)))) → TOP(mark(U41(isNat(x0), x1)))
TOP(ok(U52(x0))) → TOP(U52(active(x0)))
TOP(ok(U62(tt, x0))) → TOP(mark(s(length(x0))))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(mark(U42(x0))) → TOP(U42(proper(x0)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(ok(U42(x0))) → TOP(U42(active(x0)))
TOP(ok(U11(x0))) → TOP(U11(active(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U31(x0))) → TOP(U31(proper(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U52(x0))) → TOP(U52(proper(x0)))
TOP(mark(U41(x0, x1))) → TOP(U41(proper(x0), proper(x1)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U41(x0, x1))) → TOP(U41(active(x0), x1))
TOP(ok(U31(x0))) → TOP(U31(active(x0)))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(TOP(x₁)) = x₁
POL(U11(x₁)) = 0
POL(U21(x₁)) = 1
POL(U31(x₁)) = 0
POL(U41(x₁, x₂)) = 0
POL(U42(x₁)) = 0
POL(U51(x₁, x₂)) = 0
POL(U52(x₁)) = 0
POL(U61(x₁, x₂, x₃)) = 0
POL(U62(x₁, x₂)) = 1
POL(active(x₁)) = 0
POL(cons(x₁, x₂)) = 0
POL(isNat(x₁)) = 1 + x₁
POL(isNatIList(x₁)) = 0
POL(isNatList(x₁)) = 1
POL(length(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = x₁
POL(s(x₁)) = 1
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

U11(ok(X)) → ok(U11(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U21(ok(X)) → ok(U21(X))
U11(mark(X)) → mark(U11(X))
U31(ok(X)) → ok(U31(X))
U21(mark(X)) → mark(U21(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U31(mark(X)) → mark(U31(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
s(ok(X)) → ok(s(X))
s(mark(X)) → mark(s(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U42(mark(X)) → mark(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U52(ok(X)) → ok(U52(X))
U52(mark(X)) → mark(U52(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(zeros) → ok(zeros)

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ DependencyPairsProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                                ↳ UsableRulesReductionPairsProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

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

TOP(mark(U11(x0))) → TOP(U11(proper(x0)))
TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(ok(isNatIList(cons(x0, x1)))) → TOP(mark(U41(isNat(x0), x1)))
TOP(ok(U52(x0))) → TOP(U52(active(x0)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(ok(U62(tt, x0))) → TOP(mark(s(length(x0))))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U42(x0))) → TOP(U42(proper(x0)))
TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(ok(U42(x0))) → TOP(U42(active(x0)))
TOP(ok(U11(x0))) → TOP(U11(active(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U31(x0))) → TOP(U31(proper(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U52(x0))) → TOP(U52(proper(x0)))
TOP(mark(U41(x0, x1))) → TOP(U41(proper(x0), proper(x1)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U41(x0, x1))) → TOP(U41(active(x0), x1))
TOP(ok(U31(x0))) → TOP(U31(active(x0)))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))

The TRS R consists of the following rules:

proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
isNat(ok(X)) → ok(isNat(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
isNatList(ok(X)) → ok(isNatList(X))
U52(mark(X)) → mark(U52(X))
U52(ok(X)) → ok(U52(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
U42(mark(X)) → mark(U42(X))
U42(ok(X)) → ok(U42(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U31(mark(X)) → mark(U31(X))
U31(ok(X)) → ok(U31(X))
U21(mark(X)) → mark(U21(X))
U21(ok(X)) → ok(U21(X))
U11(mark(X)) → mark(U11(X))
U11(ok(X)) → ok(U11(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
active(zeros) → mark(cons(0, zeros))
active(U21(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))

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

The following pairs can be oriented strictly and are deleted.

TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))

The remaining pairs can at least be oriented weakly.

TOP(mark(U11(x0))) → TOP(U11(proper(x0)))
TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(ok(isNatIList(cons(x0, x1)))) → TOP(mark(U41(isNat(x0), x1)))
TOP(ok(U52(x0))) → TOP(U52(active(x0)))
TOP(ok(U62(tt, x0))) → TOP(mark(s(length(x0))))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U42(x0))) → TOP(U42(proper(x0)))
TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(ok(U42(x0))) → TOP(U42(active(x0)))
TOP(ok(U11(x0))) → TOP(U11(active(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U31(x0))) → TOP(U31(proper(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U52(x0))) → TOP(U52(proper(x0)))
TOP(mark(U41(x0, x1))) → TOP(U41(proper(x0), proper(x1)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U41(x0, x1))) → TOP(U41(active(x0), x1))
TOP(ok(U31(x0))) → TOP(U31(active(x0)))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(TOP(x₁)) = x₁
POL(U11(x₁)) = 1
POL(U21(x₁)) = 1
POL(U31(x₁)) = 1
POL(U41(x₁, x₂)) = 1
POL(U42(x₁)) = 1
POL(U51(x₁, x₂)) = 1
POL(U52(x₁)) = 1
POL(U61(x₁, x₂, x₃)) = 1
POL(U62(x₁, x₂)) = 1
POL(active(x₁)) = 0
POL(cons(x₁, x₂)) = 1
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 1
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 1
POL(mark(x₁)) = 1
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = 0
POL(s(x₁)) = 1
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

U11(ok(X)) → ok(U11(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U21(ok(X)) → ok(U21(X))
U11(mark(X)) → mark(U11(X))
U31(ok(X)) → ok(U31(X))
U21(mark(X)) → mark(U21(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U31(mark(X)) → mark(U31(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
s(ok(X)) → ok(s(X))
s(mark(X)) → mark(s(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U42(mark(X)) → mark(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U52(ok(X)) → ok(U52(X))
U52(mark(X)) → mark(U52(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ DependencyPairsProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                                ↳ UsableRulesReductionPairsProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

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

TOP(mark(U11(x0))) → TOP(U11(proper(x0)))
TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(ok(isNatIList(cons(x0, x1)))) → TOP(mark(U41(isNat(x0), x1)))
TOP(ok(U52(x0))) → TOP(U52(active(x0)))
TOP(ok(U62(tt, x0))) → TOP(mark(s(length(x0))))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(mark(U42(x0))) → TOP(U42(proper(x0)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(ok(U11(x0))) → TOP(U11(active(x0)))
TOP(ok(U42(x0))) → TOP(U42(active(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U31(x0))) → TOP(U31(proper(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U52(x0))) → TOP(U52(proper(x0)))
TOP(mark(U41(x0, x1))) → TOP(U41(proper(x0), proper(x1)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U41(x0, x1))) → TOP(U41(active(x0), x1))
TOP(ok(U31(x0))) → TOP(U31(active(x0)))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))

The TRS R consists of the following rules:

proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
isNat(ok(X)) → ok(isNat(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
isNatList(ok(X)) → ok(isNatList(X))
U52(mark(X)) → mark(U52(X))
U52(ok(X)) → ok(U52(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
U42(mark(X)) → mark(U42(X))
U42(ok(X)) → ok(U42(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U31(mark(X)) → mark(U31(X))
U31(ok(X)) → ok(U31(X))
U21(mark(X)) → mark(U21(X))
U21(ok(X)) → ok(U21(X))
U11(mark(X)) → mark(U11(X))
U11(ok(X)) → ok(U11(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
active(zeros) → mark(cons(0, zeros))
active(U21(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))

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

The following pairs can be oriented strictly and are deleted.

TOP(ok(isNatIList(cons(x0, x1)))) → TOP(mark(U41(isNat(x0), x1)))

The remaining pairs can at least be oriented weakly.

TOP(mark(U11(x0))) → TOP(U11(proper(x0)))
TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(ok(U52(x0))) → TOP(U52(active(x0)))
TOP(ok(U62(tt, x0))) → TOP(mark(s(length(x0))))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(mark(U42(x0))) → TOP(U42(proper(x0)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(ok(U11(x0))) → TOP(U11(active(x0)))
TOP(ok(U42(x0))) → TOP(U42(active(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U31(x0))) → TOP(U31(proper(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U52(x0))) → TOP(U52(proper(x0)))
TOP(mark(U41(x0, x1))) → TOP(U41(proper(x0), proper(x1)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U41(x0, x1))) → TOP(U41(active(x0), x1))
TOP(ok(U31(x0))) → TOP(U31(active(x0)))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(TOP(x₁)) = x₁
POL(U11(x₁)) = 0
POL(U21(x₁)) = 0
POL(U31(x₁)) = 0
POL(U41(x₁, x₂)) = x₁
POL(U42(x₁)) = 0
POL(U51(x₁, x₂)) = 0
POL(U52(x₁)) = 0
POL(U61(x₁, x₂, x₃)) = 0
POL(U62(x₁, x₂)) = 0
POL(active(x₁)) = x₁
POL(cons(x₁, x₂)) = 1
POL(isNat(x₁)) = 1
POL(isNatIList(x₁)) = 1 + x₁
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = x₁
POL(s(x₁)) = 0
POL(tt) = 0
POL(zeros) = 1

The following usable rules [17] were oriented:

U11(ok(X)) → ok(U11(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U21(ok(X)) → ok(U21(X))
U11(mark(X)) → mark(U11(X))
U31(ok(X)) → ok(U31(X))
U21(mark(X)) → mark(U21(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U31(mark(X)) → mark(U31(X))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U21(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
active(zeros) → mark(cons(0, zeros))
active(U11(X)) → U11(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(0)) → mark(tt)
active(U62(tt, L)) → mark(s(length(L)))
active(U62(X1, X2)) → U62(active(X1), X2)
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U52(X)) → U52(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U31(X)) → U31(active(X))
active(U21(X)) → U21(active(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
s(ok(X)) → ok(s(X))
s(mark(X)) → mark(s(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U42(mark(X)) → mark(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U52(ok(X)) → ok(U52(X))
U52(mark(X)) → mark(U52(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
active(s(X)) → s(active(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
active(length(X)) → length(active(X))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(zeros) → ok(zeros)

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ DependencyPairsProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                                ↳ UsableRulesReductionPairsProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

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

TOP(mark(U11(x0))) → TOP(U11(proper(x0)))
TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(ok(U52(x0))) → TOP(U52(active(x0)))
TOP(ok(U62(tt, x0))) → TOP(mark(s(length(x0))))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U42(x0))) → TOP(U42(proper(x0)))
TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(ok(U42(x0))) → TOP(U42(active(x0)))
TOP(ok(U11(x0))) → TOP(U11(active(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U31(x0))) → TOP(U31(proper(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U52(x0))) → TOP(U52(proper(x0)))
TOP(mark(U41(x0, x1))) → TOP(U41(proper(x0), proper(x1)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U41(x0, x1))) → TOP(U41(active(x0), x1))
TOP(ok(U31(x0))) → TOP(U31(active(x0)))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))

The TRS R consists of the following rules:

proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
isNat(ok(X)) → ok(isNat(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
isNatList(ok(X)) → ok(isNatList(X))
U52(mark(X)) → mark(U52(X))
U52(ok(X)) → ok(U52(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
U42(mark(X)) → mark(U42(X))
U42(ok(X)) → ok(U42(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U31(mark(X)) → mark(U31(X))
U31(ok(X)) → ok(U31(X))
U21(mark(X)) → mark(U21(X))
U21(ok(X)) → ok(U21(X))
U11(mark(X)) → mark(U11(X))
U11(ok(X)) → ok(U11(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
active(zeros) → mark(cons(0, zeros))
active(U21(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))

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

The following pairs can be oriented strictly and are deleted.

TOP(ok(U62(tt, x0))) → TOP(mark(s(length(x0))))

The remaining pairs can at least be oriented weakly.

TOP(mark(U11(x0))) → TOP(U11(proper(x0)))
TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(ok(U52(x0))) → TOP(U52(active(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(U42(x0))) → TOP(U42(proper(x0)))
TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(ok(U42(x0))) → TOP(U42(active(x0)))
TOP(ok(U11(x0))) → TOP(U11(active(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U31(x0))) → TOP(U31(proper(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U52(x0))) → TOP(U52(proper(x0)))
TOP(mark(U41(x0, x1))) → TOP(U41(proper(x0), proper(x1)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U41(x0, x1))) → TOP(U41(active(x0), x1))
TOP(ok(U31(x0))) → TOP(U31(active(x0)))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(TOP(x₁)) = x₁
POL(U11(x₁)) = 0
POL(U21(x₁)) = x₁
POL(U31(x₁)) = 0
POL(U41(x₁, x₂)) = 0
POL(U42(x₁)) = x₁
POL(U51(x₁, x₂)) = 0
POL(U52(x₁)) = x₁
POL(U61(x₁, x₂, x₃)) = x₁
POL(U62(x₁, x₂)) = x₁
POL(active(x₁)) = x₁
POL(cons(x₁, x₂)) = 0
POL(isNat(x₁)) = 1
POL(isNatIList(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = x₁
POL(s(x₁)) = 0
POL(tt) = 1
POL(zeros) = 0

The following usable rules [17] were oriented:

U11(ok(X)) → ok(U11(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U21(ok(X)) → ok(U21(X))
U11(mark(X)) → mark(U11(X))
U31(ok(X)) → ok(U31(X))
U21(mark(X)) → mark(U21(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U31(mark(X)) → mark(U31(X))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U21(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
active(zeros) → mark(cons(0, zeros))
active(U11(X)) → U11(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(0)) → mark(tt)
active(U62(tt, L)) → mark(s(length(L)))
active(U62(X1, X2)) → U62(active(X1), X2)
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U52(X)) → U52(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U31(X)) → U31(active(X))
active(U21(X)) → U21(active(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
s(ok(X)) → ok(s(X))
s(mark(X)) → mark(s(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U42(mark(X)) → mark(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U52(ok(X)) → ok(U52(X))
U52(mark(X)) → mark(U52(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))
active(s(X)) → s(active(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
active(length(X)) → length(active(X))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(zeros) → ok(zeros)

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ DependencyPairsProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                                ↳ UsableRulesReductionPairsProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

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

TOP(mark(U11(x0))) → TOP(U11(proper(x0)))
TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(ok(U52(x0))) → TOP(U52(active(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(mark(U42(x0))) → TOP(U42(proper(x0)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(ok(U11(x0))) → TOP(U11(active(x0)))
TOP(ok(U42(x0))) → TOP(U42(active(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U31(x0))) → TOP(U31(proper(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U52(x0))) → TOP(U52(proper(x0)))
TOP(mark(U41(x0, x1))) → TOP(U41(proper(x0), proper(x1)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U41(x0, x1))) → TOP(U41(active(x0), x1))
TOP(ok(U31(x0))) → TOP(U31(active(x0)))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))

The TRS R consists of the following rules:

proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
isNat(ok(X)) → ok(isNat(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
isNatList(ok(X)) → ok(isNatList(X))
U52(mark(X)) → mark(U52(X))
U52(ok(X)) → ok(U52(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
U42(mark(X)) → mark(U42(X))
U42(ok(X)) → ok(U42(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U31(mark(X)) → mark(U31(X))
U31(ok(X)) → ok(U31(X))
U21(mark(X)) → mark(U21(X))
U21(ok(X)) → ok(U21(X))
U11(mark(X)) → mark(U11(X))
U11(ok(X)) → ok(U11(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
active(zeros) → mark(cons(0, zeros))
active(U21(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))

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

The following pairs can be oriented strictly and are deleted.

TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))

The remaining pairs can at least be oriented weakly.

TOP(mark(U11(x0))) → TOP(U11(proper(x0)))
TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(ok(U52(x0))) → TOP(U52(active(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(mark(U42(x0))) → TOP(U42(proper(x0)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(ok(U11(x0))) → TOP(U11(active(x0)))
TOP(ok(U42(x0))) → TOP(U42(active(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(U31(x0))) → TOP(U31(proper(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U52(x0))) → TOP(U52(proper(x0)))
TOP(mark(U41(x0, x1))) → TOP(U41(proper(x0), proper(x1)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U41(x0, x1))) → TOP(U41(active(x0), x1))
TOP(ok(U31(x0))) → TOP(U31(active(x0)))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(TOP(x₁)) = x₁
POL(U11(x₁)) = 1
POL(U21(x₁)) = 1
POL(U31(x₁)) = 1
POL(U41(x₁, x₂)) = 1
POL(U42(x₁)) = 1
POL(U51(x₁, x₂)) = 1
POL(U52(x₁)) = 1
POL(U61(x₁, x₂, x₃)) = 1
POL(U62(x₁, x₂)) = 1
POL(active(x₁)) = 0
POL(cons(x₁, x₂)) = 1
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 1
POL(mark(x₁)) = 1
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = 0
POL(s(x₁)) = 1
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

U11(ok(X)) → ok(U11(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U21(ok(X)) → ok(U21(X))
U11(mark(X)) → mark(U11(X))
U31(ok(X)) → ok(U31(X))
U21(mark(X)) → mark(U21(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U31(mark(X)) → mark(U31(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
s(ok(X)) → ok(s(X))
s(mark(X)) → mark(s(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U42(mark(X)) → mark(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U52(ok(X)) → ok(U52(X))
U52(mark(X)) → mark(U52(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U51(mark(X1), X2) → mark(U51(X1, X2))

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ DependencyPairsProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ AND

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                              ↳ QDP

                                ↳ UsableRulesReductionPairsProof

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ QDPOrderProof

                                                                              ↳ QDP

                                                                                ↳ QDPOrderProof

                                                                                  ↳ QDP

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

                                                                                          ↳ QDP

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

TOP(mark(U11(x0))) → TOP(U11(proper(x0)))
TOP(ok(U62(x0, x1))) → TOP(U62(active(x0), x1))
TOP(mark(U51(x0, x1))) → TOP(U51(proper(x0), proper(x1)))
TOP(ok(U11(x0))) → TOP(U11(active(x0)))
TOP(ok(U42(x0))) → TOP(U42(active(x0)))
TOP(mark(U62(x0, x1))) → TOP(U62(proper(x0), proper(x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U31(x0))) → TOP(U31(proper(x0)))
TOP(mark(U52(x0))) → TOP(U52(proper(x0)))
TOP(ok(U52(x0))) → TOP(U52(active(x0)))
TOP(mark(U41(x0, x1))) → TOP(U41(proper(x0), proper(x1)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U61(x0, x1, x2))) → TOP(U61(active(x0), x1, x2))
TOP(ok(U41(x0, x1))) → TOP(U41(active(x0), x1))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(U31(x0))) → TOP(U31(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(U51(x0, x1))) → TOP(U51(active(x0), x1))
TOP(mark(U42(x0))) → TOP(U42(proper(x0)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(U61(x0, x1, x2))) → TOP(U61(proper(x0), proper(x1), proper(x2)))

The TRS R consists of the following rules:

proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) → U62(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
length(mark(X)) → mark(length(X))
length(ok(X)) → ok(length(X))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
isNat(ok(X)) → ok(isNat(X))
U62(mark(X1), X2) → mark(U62(X1, X2))
U62(ok(X1), ok(X2)) → ok(U62(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
isNatList(ok(X)) → ok(isNatList(X))
U52(mark(X)) → mark(U52(X))
U52(ok(X)) → ok(U52(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
isNatIList(ok(X)) → ok(isNatIList(X))
U42(mark(X)) → mark(U42(X))
U42(ok(X)) → ok(U42(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U31(mark(X)) → mark(U31(X))
U31(ok(X)) → ok(U31(X))
U21(mark(X)) → mark(U21(X))
U21(ok(X)) → ok(U21(X))
U11(mark(X)) → mark(U11(X))
U11(ok(X)) → ok(U11(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
active(zeros) → mark(cons(0, zeros))
active(U21(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, L, N)) → mark(U62(isNat(N), L))
active(U62(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(length(cons(N, L))) → mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2)) → U62(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))

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