Termination of the following Term Rewriting System could be disproven:

Generalized rewrite system (where rules with free variables on rhs are allowed):
The TRS R consists of the following rules:

U72(tt) → s(length(L))
U93(tt) → cons(N)
zeroscons(0)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt) → U42(isNatIList)
U42(tt) → tt
U51(tt) → U52(isNatList)
U52(tt) → tt
U61(tt) → U62(isNatIList)
U62(tt) → tt
U71(tt) → U72(isNat)
U81(tt) → nil
U91(tt) → U92(isNat)
U92(tt) → U93(isNat)
isNattt
isNatU11(isNatList)
isNatU21(isNat)
isNatIListU31(isNatList)
isNatIListtt
isNatIListU41(isNat)
isNatListtt
isNatListU51(isNat)
isNatListU61(isNat)
length(nil) → 0
length(cons(N)) → U71(isNatList)
take(0, IL) → U81(isNatIList)
take(s(M), cons(N)) → U91(isNatIList)



GTRS
  ↳ CritRuleProof

Generalized rewrite system (where rules with free variables on rhs are allowed):
The TRS R consists of the following rules:

U72(tt) → s(length(L))
U93(tt) → cons(N)
zeroscons(0)
U11(tt) → tt
U21(tt) → tt
U31(tt) → tt
U41(tt) → U42(isNatIList)
U42(tt) → tt
U51(tt) → U52(isNatList)
U52(tt) → tt
U61(tt) → U62(isNatIList)
U62(tt) → tt
U71(tt) → U72(isNat)
U81(tt) → nil
U91(tt) → U92(isNat)
U92(tt) → U93(isNat)
isNattt
isNatU11(isNatList)
isNatU21(isNat)
isNatIListU31(isNatList)
isNatIListtt
isNatIListU41(isNat)
isNatListtt
isNatListU51(isNat)
isNatListU61(isNat)
length(nil) → 0
length(cons(N)) → U71(isNatList)
take(0, IL) → U81(isNatIList)
take(s(M), cons(N)) → U91(isNatIList)


The rule U72(tt) → s(length(L)) contains free variables in its right-hand side. Hence the TRS is not-terminating.