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:

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
isNat → tt
isNat → U11(and(isNatKind))
isNat → U21(isNatKind)
isNat → U31(and(isNatKind))
isNatKind → tt
isNatKind → and(isNatKind)
isNatKind → isNatKind
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)))

Innermost Strategy.


↳ 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
isNat → tt
isNat → U11(and(isNatKind))
isNat → U21(isNatKind)
isNat → U31(and(isNatKind))
isNatKind → tt
isNatKind → and(isNatKind)
isNatKind → isNatKind
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)))

Innermost Strategy.

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