The TRS could not be proven terminating. The proof attempt took 60004 ms.
Problem 1 was processed with processor DependencyGraph (1188ms). | – Problem 2 remains open; application of the following processors failed [SubtermCriterion (2ms), DependencyGraph (843ms), PolynomialLinearRange4iUR (timeout), DependencyGraph (611ms), PolynomialLinearRange8NegiUR (timeout), DependencyGraph (626ms), ReductionPairSAT (timeout)].
| U51#(tt, M, N) | → | activate#(N) | plus#(N, 0) | → | U41#(isNat(N), N) | |
| U41#(tt, N) | → | activate#(N) | activate#(n__s(X)) | → | activate#(X) | |
| activate#(n__plus(X1, X2)) | → | plus#(activate(X1), activate(X2)) | isNat#(n__x(V1, V2)) | → | activate#(V2) | |
| activate#(n__x(X1, X2)) | → | activate#(X1) | U52#(tt, M, N) | → | plus#(activate(N), activate(M)) | |
| isNat#(n__plus(V1, V2)) | → | isNat#(activate(V1)) | U71#(tt, M, N) | → | U72#(isNat(activate(N)), activate(M), activate(N)) | |
| plus#(N, s(M)) | → | U51#(isNat(M), M, N) | isNat#(n__plus(V1, V2)) | → | U11#(isNat(activate(V1)), activate(V2)) | |
| isNat#(n__plus(V1, V2)) | → | activate#(V2) | plus#(N, 0) | → | isNat#(N) | |
| U72#(tt, M, N) | → | x#(activate(N), activate(M)) | x#(N, 0) | → | isNat#(N) | |
| activate#(n__x(X1, X2)) | → | x#(activate(X1), activate(X2)) | activate#(n__plus(X1, X2)) | → | activate#(X2) | |
| activate#(n__x(X1, X2)) | → | activate#(X2) | U11#(tt, V2) | → | activate#(V2) | |
| isNat#(n__x(V1, V2)) | → | U31#(isNat(activate(V1)), activate(V2)) | U51#(tt, M, N) | → | U52#(isNat(activate(N)), activate(M), activate(N)) | |
| isNat#(n__x(V1, V2)) | → | activate#(V1) | isNat#(n__s(V1)) | → | activate#(V1) | |
| activate#(n__plus(X1, X2)) | → | activate#(X1) | x#(N, s(M)) | → | isNat#(M) | |
| U52#(tt, M, N) | → | activate#(N) | U72#(tt, M, N) | → | plus#(x(activate(N), activate(M)), activate(N)) | |
| U31#(tt, V2) | → | isNat#(activate(V2)) | U11#(tt, V2) | → | isNat#(activate(V2)) | |
| U52#(tt, M, N) | → | activate#(M) | U71#(tt, M, N) | → | activate#(N) | |
| x#(N, s(M)) | → | U71#(isNat(M), M, N) | plus#(N, s(M)) | → | isNat#(M) | |
| U71#(tt, M, N) | → | isNat#(activate(N)) | U51#(tt, M, N) | → | isNat#(activate(N)) | |
| isNat#(n__s(V1)) | → | isNat#(activate(V1)) | U31#(tt, V2) | → | activate#(V2) | |
| isNat#(n__plus(V1, V2)) | → | activate#(V1) | U51#(tt, M, N) | → | activate#(M) | |
| U72#(tt, M, N) | → | activate#(N) | U71#(tt, M, N) | → | activate#(M) | |
| U72#(tt, M, N) | → | activate#(M) | isNat#(n__x(V1, V2)) | → | isNat#(activate(V1)) |
| U11(tt, V2) | → | U12(isNat(activate(V2))) | U12(tt) | → | tt | |
| U21(tt) | → | tt | U31(tt, V2) | → | U32(isNat(activate(V2))) | |
| U32(tt) | → | tt | U41(tt, N) | → | activate(N) | |
| U51(tt, M, N) | → | U52(isNat(activate(N)), activate(M), activate(N)) | U52(tt, M, N) | → | s(plus(activate(N), activate(M))) | |
| U61(tt) | → | 0 | U71(tt, M, N) | → | U72(isNat(activate(N)), activate(M), activate(N)) | |
| U72(tt, M, N) | → | plus(x(activate(N), activate(M)), activate(N)) | isNat(n__0) | → | tt | |
| isNat(n__plus(V1, V2)) | → | U11(isNat(activate(V1)), activate(V2)) | isNat(n__s(V1)) | → | U21(isNat(activate(V1))) | |
| isNat(n__x(V1, V2)) | → | U31(isNat(activate(V1)), activate(V2)) | plus(N, 0) | → | U41(isNat(N), N) | |
| plus(N, s(M)) | → | U51(isNat(M), M, N) | x(N, 0) | → | U61(isNat(N)) | |
| x(N, s(M)) | → | U71(isNat(M), M, N) | 0 | → | n__0 | |
| plus(X1, X2) | → | n__plus(X1, X2) | s(X) | → | n__s(X) | |
| x(X1, X2) | → | n__x(X1, X2) | activate(n__0) | → | 0 | |
| activate(n__plus(X1, X2)) | → | plus(activate(X1), activate(X2)) | activate(n__s(X)) | → | s(activate(X)) | |
| activate(n__x(X1, X2)) | → | x(activate(X1), activate(X2)) | activate(X) | → | X |
Termination of terms over the following signature is verified: plus, n__plus, U71, n__s, activate, isNat, U72, n__0, U61, 0, U51, s, tt, U41, U52, U11, U12, U31, U32, U21, n__x, x
| U61#(tt) | → | 0# | plus#(N, 0) | → | U41#(isNat(N), N) | |
| U41#(tt, N) | → | activate#(N) | isNat#(n__x(V1, V2)) | → | activate#(V2) | |
| activate#(n__x(X1, X2)) | → | activate#(X1) | U71#(tt, M, N) | → | U72#(isNat(activate(N)), activate(M), activate(N)) | |
| plus#(N, s(M)) | → | U51#(isNat(M), M, N) | U11#(tt, V2) | → | U12#(isNat(activate(V2))) | |
| U72#(tt, M, N) | → | x#(activate(N), activate(M)) | activate#(n__x(X1, X2)) | → | activate#(X2) | |
| isNat#(n__s(V1)) | → | activate#(V1) | U31#(tt, V2) | → | U32#(isNat(activate(V2))) | |
| x#(N, s(M)) | → | isNat#(M) | U72#(tt, M, N) | → | plus#(x(activate(N), activate(M)), activate(N)) | |
| U31#(tt, V2) | → | isNat#(activate(V2)) | U71#(tt, M, N) | → | activate#(N) | |
| x#(N, s(M)) | → | U71#(isNat(M), M, N) | isNat#(n__s(V1)) | → | U21#(isNat(activate(V1))) | |
| U51#(tt, M, N) | → | isNat#(activate(N)) | activate#(n__0) | → | 0# | |
| isNat#(n__plus(V1, V2)) | → | activate#(V1) | U51#(tt, M, N) | → | activate#(M) | |
| U72#(tt, M, N) | → | activate#(M) | isNat#(n__x(V1, V2)) | → | isNat#(activate(V1)) | |
| U51#(tt, M, N) | → | activate#(N) | activate#(n__plus(X1, X2)) | → | plus#(activate(X1), activate(X2)) | |
| activate#(n__s(X)) | → | activate#(X) | U52#(tt, M, N) | → | plus#(activate(N), activate(M)) | |
| isNat#(n__plus(V1, V2)) | → | isNat#(activate(V1)) | U52#(tt, M, N) | → | s#(plus(activate(N), activate(M))) | |
| isNat#(n__plus(V1, V2)) | → | U11#(isNat(activate(V1)), activate(V2)) | isNat#(n__plus(V1, V2)) | → | activate#(V2) | |
| plus#(N, 0) | → | isNat#(N) | activate#(n__x(X1, X2)) | → | x#(activate(X1), activate(X2)) | |
| x#(N, 0) | → | isNat#(N) | activate#(n__plus(X1, X2)) | → | activate#(X2) | |
| U11#(tt, V2) | → | activate#(V2) | isNat#(n__x(V1, V2)) | → | U31#(isNat(activate(V1)), activate(V2)) | |
| U51#(tt, M, N) | → | U52#(isNat(activate(N)), activate(M), activate(N)) | isNat#(n__x(V1, V2)) | → | activate#(V1) | |
| activate#(n__plus(X1, X2)) | → | activate#(X1) | U52#(tt, M, N) | → | activate#(N) | |
| x#(N, 0) | → | U61#(isNat(N)) | U52#(tt, M, N) | → | activate#(M) | |
| U11#(tt, V2) | → | isNat#(activate(V2)) | plus#(N, s(M)) | → | isNat#(M) | |
| U71#(tt, M, N) | → | isNat#(activate(N)) | isNat#(n__s(V1)) | → | isNat#(activate(V1)) | |
| U31#(tt, V2) | → | activate#(V2) | activate#(n__s(X)) | → | s#(activate(X)) | |
| U72#(tt, M, N) | → | activate#(N) | U71#(tt, M, N) | → | activate#(M) |
| U11(tt, V2) | → | U12(isNat(activate(V2))) | U12(tt) | → | tt | |
| U21(tt) | → | tt | U31(tt, V2) | → | U32(isNat(activate(V2))) | |
| U32(tt) | → | tt | U41(tt, N) | → | activate(N) | |
| U51(tt, M, N) | → | U52(isNat(activate(N)), activate(M), activate(N)) | U52(tt, M, N) | → | s(plus(activate(N), activate(M))) | |
| U61(tt) | → | 0 | U71(tt, M, N) | → | U72(isNat(activate(N)), activate(M), activate(N)) | |
| U72(tt, M, N) | → | plus(x(activate(N), activate(M)), activate(N)) | isNat(n__0) | → | tt | |
| isNat(n__plus(V1, V2)) | → | U11(isNat(activate(V1)), activate(V2)) | isNat(n__s(V1)) | → | U21(isNat(activate(V1))) | |
| isNat(n__x(V1, V2)) | → | U31(isNat(activate(V1)), activate(V2)) | plus(N, 0) | → | U41(isNat(N), N) | |
| plus(N, s(M)) | → | U51(isNat(M), M, N) | x(N, 0) | → | U61(isNat(N)) | |
| x(N, s(M)) | → | U71(isNat(M), M, N) | 0 | → | n__0 | |
| plus(X1, X2) | → | n__plus(X1, X2) | s(X) | → | n__s(X) | |
| x(X1, X2) | → | n__x(X1, X2) | activate(n__0) | → | 0 | |
| activate(n__plus(X1, X2)) | → | plus(activate(X1), activate(X2)) | activate(n__s(X)) | → | s(activate(X)) | |
| activate(n__x(X1, X2)) | → | x(activate(X1), activate(X2)) | activate(X) | → | X |
Termination of terms over the following signature is verified: plus, n__plus, U71, n__s, activate, isNat, U72, n__0, U61, 0, U51, s, U41, tt, U52, U11, U12, U31, U32, U21, x, n__x
| U51#(tt, M, N) → activate#(N) | plus#(N, 0) → U41#(isNat(N), N) |
| U41#(tt, N) → activate#(N) | activate#(n__plus(X1, X2)) → plus#(activate(X1), activate(X2)) |
| activate#(n__s(X)) → activate#(X) | isNat#(n__x(V1, V2)) → activate#(V2) |
| activate#(n__x(X1, X2)) → activate#(X1) | isNat#(n__plus(V1, V2)) → isNat#(activate(V1)) |
| U52#(tt, M, N) → plus#(activate(N), activate(M)) | plus#(N, s(M)) → U51#(isNat(M), M, N) |
| U71#(tt, M, N) → U72#(isNat(activate(N)), activate(M), activate(N)) | isNat#(n__plus(V1, V2)) → U11#(isNat(activate(V1)), activate(V2)) |
| plus#(N, 0) → isNat#(N) | isNat#(n__plus(V1, V2)) → activate#(V2) |
| activate#(n__x(X1, X2)) → x#(activate(X1), activate(X2)) | x#(N, 0) → isNat#(N) |
| U72#(tt, M, N) → x#(activate(N), activate(M)) | activate#(n__plus(X1, X2)) → activate#(X2) |
| activate#(n__x(X1, X2)) → activate#(X2) | U11#(tt, V2) → activate#(V2) |
| isNat#(n__x(V1, V2)) → U31#(isNat(activate(V1)), activate(V2)) | U51#(tt, M, N) → U52#(isNat(activate(N)), activate(M), activate(N)) |
| isNat#(n__x(V1, V2)) → activate#(V1) | isNat#(n__s(V1)) → activate#(V1) |
| activate#(n__plus(X1, X2)) → activate#(X1) | x#(N, s(M)) → isNat#(M) |
| U52#(tt, M, N) → activate#(N) | U72#(tt, M, N) → plus#(x(activate(N), activate(M)), activate(N)) |
| U31#(tt, V2) → isNat#(activate(V2)) | U52#(tt, M, N) → activate#(M) |
| U11#(tt, V2) → isNat#(activate(V2)) | U71#(tt, M, N) → activate#(N) |
| plus#(N, s(M)) → isNat#(M) | x#(N, s(M)) → U71#(isNat(M), M, N) |
| U71#(tt, M, N) → isNat#(activate(N)) | U51#(tt, M, N) → isNat#(activate(N)) |
| isNat#(n__s(V1)) → isNat#(activate(V1)) | U31#(tt, V2) → activate#(V2) |
| isNat#(n__plus(V1, V2)) → activate#(V1) | U51#(tt, M, N) → activate#(M) |
| U72#(tt, M, N) → activate#(N) | U71#(tt, M, N) → activate#(M) |
| U72#(tt, M, N) → activate#(M) | isNat#(n__x(V1, V2)) → isNat#(activate(V1)) |