Innermost 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:

U114(tt) → s(length(L))
U136(tt) → cons(N)
zeros → cons(0)
U101(tt) → U102(isNatKind)
U102(tt) → U103(isNatIListKind)
U103(tt) → U104(isNatIListKind)
U104(tt) → U105(isNat)
U105(tt) → U106(isNatIList)
U106(tt) → tt
U11(tt) → U12(isNatIListKind)
U111(tt) → U112(isNatIListKind)
U112(tt) → U113(isNat)
U113(tt) → U114(isNatKind)
U12(tt) → U13(isNatList)
U121(tt) → U122(isNatIListKind)
U122(tt) → nil
U13(tt) → tt
U131(tt) → U132(isNatIListKind)
U132(tt) → U133(isNat)
U133(tt) → U134(isNatKind)
U134(tt) → U135(isNat)
U135(tt) → U136(isNatKind)
U21(tt) → U22(isNatKind)
U22(tt) → U23(isNat)
U23(tt) → tt
U31(tt) → U32(isNatIListKind)
U32(tt) → U33(isNatList)
U33(tt) → tt
U41(tt) → U42(isNatKind)
U42(tt) → U43(isNatIListKind)
U43(tt) → U44(isNatIListKind)
U44(tt) → U45(isNat)
U45(tt) → U46(isNatIList)
U46(tt) → tt
U51(tt) → U52(isNatIListKind)
U52(tt) → tt
U61(tt) → U62(isNatIListKind)
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt) → U92(isNatKind)
U92(tt) → U93(isNatIListKind)
U93(tt) → U94(isNatIListKind)
U94(tt) → U95(isNat)
U95(tt) → U96(isNatList)
U96(tt) → tt
isNat → tt
isNat → U11(isNatIListKind)
isNat → U21(isNatKind)
isNatIList → U31(isNatIListKind)
isNatIList → tt
isNatIList → U41(isNatKind)
isNatIListKind → tt
isNatIListKind → U51(isNatKind)
isNatIListKind → U61(isNatKind)
isNatKind → tt
isNatKind → U71(isNatIListKind)
isNatKind → U81(isNatKind)
isNatList → tt
isNatList → U91(isNatKind)
isNatList → U101(isNatKind)
length(nil) → 0
length(cons(N)) → U111(isNatList)
take(0, IL) → U121(isNatIList)
take(s(M), cons(N)) → U131(isNatIList)

Innermost Strategy.


↳ GTRS
  ↳ CritRuleProof

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

U114(tt) → s(length(L))
U136(tt) → cons(N)
zeros → cons(0)
U101(tt) → U102(isNatKind)
U102(tt) → U103(isNatIListKind)
U103(tt) → U104(isNatIListKind)
U104(tt) → U105(isNat)
U105(tt) → U106(isNatIList)
U106(tt) → tt
U11(tt) → U12(isNatIListKind)
U111(tt) → U112(isNatIListKind)
U112(tt) → U113(isNat)
U113(tt) → U114(isNatKind)
U12(tt) → U13(isNatList)
U121(tt) → U122(isNatIListKind)
U122(tt) → nil
U13(tt) → tt
U131(tt) → U132(isNatIListKind)
U132(tt) → U133(isNat)
U133(tt) → U134(isNatKind)
U134(tt) → U135(isNat)
U135(tt) → U136(isNatKind)
U21(tt) → U22(isNatKind)
U22(tt) → U23(isNat)
U23(tt) → tt
U31(tt) → U32(isNatIListKind)
U32(tt) → U33(isNatList)
U33(tt) → tt
U41(tt) → U42(isNatKind)
U42(tt) → U43(isNatIListKind)
U43(tt) → U44(isNatIListKind)
U44(tt) → U45(isNat)
U45(tt) → U46(isNatIList)
U46(tt) → tt
U51(tt) → U52(isNatIListKind)
U52(tt) → tt
U61(tt) → U62(isNatIListKind)
U62(tt) → tt
U71(tt) → tt
U81(tt) → tt
U91(tt) → U92(isNatKind)
U92(tt) → U93(isNatIListKind)
U93(tt) → U94(isNatIListKind)
U94(tt) → U95(isNat)
U95(tt) → U96(isNatList)
U96(tt) → tt
isNat → tt
isNat → U11(isNatIListKind)
isNat → U21(isNatKind)
isNatIList → U31(isNatIListKind)
isNatIList → tt
isNatIList → U41(isNatKind)
isNatIListKind → tt
isNatIListKind → U51(isNatKind)
isNatIListKind → U61(isNatKind)
isNatKind → tt
isNatKind → U71(isNatIListKind)
isNatKind → U81(isNatKind)
isNatList → tt
isNatList → U91(isNatKind)
isNatList → U101(isNatKind)
length(nil) → 0
length(cons(N)) → U111(isNatList)
take(0, IL) → U121(isNatIList)
take(s(M), cons(N)) → U131(isNatIList)

Innermost Strategy.

The rule U114(tt) → s(length(L)) contains free variables in its right-hand side. Hence the TRS is not-terminating.