(0) Obligation:
Generalized rewrite system (where rules with free variables on rhs are allowed):
The TRS R consists of the following rules:
U11(tt) → s(length(L))
U31(tt) → cons(N)
and(tt) → X
zeros → cons(0)
U21(tt) → nil
isNat → tt
isNat → isNatList
isNat → isNat
isNatIList → isNatList
isNatIList → tt
isNatIList → and(isNat)
isNatList → tt
isNatList → and(isNat)
length(nil) → 0
length(cons(N)) → U11(and(isNatList))
take(0, IL) → U21(isNatIList)
take(s(M), cons(N)) → U31(and(isNatIList))
(1) CritRuleProof (EQUIVALENT transformation)
The rule U11(tt) → s(length(L)) contains free variables in its right-hand side. Hence the TRS is not-terminating.
(2) NO