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:

U41(tt) → N
U51(tt) → s(plus(N, M))
U71(tt) → plus(x(N, M), N)
and(tt) → X
U11(tt) → U12(isNat)
U12(tt) → U13(isNat)
U13(tt) → tt
U21(tt) → U22(isNat)
U22(tt) → tt
U31(tt) → U32(isNat)
U32(tt) → U33(isNat)
U33(tt) → tt
U61(tt) → 0
isNattt
isNatU11(and(isNatKind))
isNatU21(isNatKind)
isNatU31(and(isNatKind))
isNatKindtt
isNatKindand(isNatKind)
isNatKindisNatKind
plus(N, 0) → U41(and(isNat))
plus(N, s(M)) → U51(and(and(isNat)))
x(N, 0) → U61(and(isNat))
x(N, s(M)) → U71(and(and(isNat)))



GTRS
  ↳ CritRuleProof

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

U41(tt) → N
U51(tt) → s(plus(N, M))
U71(tt) → plus(x(N, M), N)
and(tt) → X
U11(tt) → U12(isNat)
U12(tt) → U13(isNat)
U13(tt) → tt
U21(tt) → U22(isNat)
U22(tt) → tt
U31(tt) → U32(isNat)
U32(tt) → U33(isNat)
U33(tt) → tt
U61(tt) → 0
isNattt
isNatU11(and(isNatKind))
isNatU21(isNatKind)
isNatU31(and(isNatKind))
isNatKindtt
isNatKindand(isNatKind)
isNatKindisNatKind
plus(N, 0) → U41(and(isNat))
plus(N, s(M)) → U51(and(and(isNat)))
x(N, 0) → U61(and(isNat))
x(N, s(M)) → U71(and(and(isNat)))


The rule U41(tt) → N contains free variables in its right-hand side. Hence the TRS is not-terminating.