The TRS could not be proven terminating. The proof attempt took 60007 ms.
Problem 1 was processed with processor DependencyGraph (1103ms). | – Problem 2 remains open; application of the following processors failed [SubtermCriterion (1ms), DependencyGraph (901ms), PolynomialLinearRange4iUR (timeout), DependencyGraph (708ms), PolynomialLinearRange8NegiUR (timeout), DependencyGraph (702ms), ReductionPairSAT (timeout)].
| mark#(U11(X1, X2)) | → | mark#(X1) | a__isNat#(plus(V1, V2)) | → | a__and#(a__isNat(V1), isNat(V2)) | |
| mark#(isNat(X)) | → | a__isNat#(X) | mark#(x(X1, X2)) | → | a__x#(mark(X1), mark(X2)) | |
| a__isNat#(x(V1, V2)) | → | a__and#(a__isNat(V1), isNat(V2)) | a__x#(N, s(M)) | → | a__and#(a__isNat(M), isNat(N)) | |
| mark#(U41(X1, X2, X3)) | → | a__U41#(mark(X1), X2, X3) | a__isNat#(s(V1)) | → | a__isNat#(V1) | |
| a__plus#(N, 0) | → | a__U11#(a__isNat(N), N) | a__U41#(tt, M, N) | → | mark#(M) | |
| mark#(plus(X1, X2)) | → | mark#(X2) | a__and#(tt, X) | → | mark#(X) | |
| a__U41#(tt, M, N) | → | a__plus#(a__x(mark(N), mark(M)), mark(N)) | a__isNat#(plus(V1, V2)) | → | a__isNat#(V1) | |
| a__plus#(N, s(M)) | → | a__isNat#(M) | mark#(s(X)) | → | mark#(X) | |
| a__x#(N, 0) | → | a__isNat#(N) | a__plus#(N, s(M)) | → | a__and#(a__isNat(M), isNat(N)) | |
| a__U21#(tt, M, N) | → | mark#(M) | a__U11#(tt, N) | → | mark#(N) | |
| a__U21#(tt, M, N) | → | mark#(N) | mark#(plus(X1, X2)) | → | a__plus#(mark(X1), mark(X2)) | |
| a__U41#(tt, M, N) | → | a__x#(mark(N), mark(M)) | a__x#(N, s(M)) | → | a__U41#(a__and(a__isNat(M), isNat(N)), M, N) | |
| a__U21#(tt, M, N) | → | a__plus#(mark(N), mark(M)) | mark#(U21(X1, X2, X3)) | → | a__U21#(mark(X1), X2, X3) | |
| mark#(plus(X1, X2)) | → | mark#(X1) | mark#(and(X1, X2)) | → | mark#(X1) | |
| a__plus#(N, s(M)) | → | a__U21#(a__and(a__isNat(M), isNat(N)), M, N) | mark#(x(X1, X2)) | → | mark#(X2) | |
| a__x#(N, s(M)) | → | a__isNat#(M) | mark#(and(X1, X2)) | → | a__and#(mark(X1), X2) | |
| mark#(U41(X1, X2, X3)) | → | mark#(X1) | a__U41#(tt, M, N) | → | mark#(N) | |
| a__plus#(N, 0) | → | a__isNat#(N) | mark#(U31(X)) | → | mark#(X) | |
| mark#(U11(X1, X2)) | → | a__U11#(mark(X1), X2) | mark#(U21(X1, X2, X3)) | → | mark#(X1) | |
| a__isNat#(x(V1, V2)) | → | a__isNat#(V1) | mark#(x(X1, X2)) | → | mark#(X1) |
| a__U11(tt, N) | → | mark(N) | a__U21(tt, M, N) | → | s(a__plus(mark(N), mark(M))) | |
| a__U31(tt) | → | 0 | a__U41(tt, M, N) | → | a__plus(a__x(mark(N), mark(M)), mark(N)) | |
| a__and(tt, X) | → | mark(X) | a__isNat(0) | → | tt | |
| a__isNat(plus(V1, V2)) | → | a__and(a__isNat(V1), isNat(V2)) | a__isNat(s(V1)) | → | a__isNat(V1) | |
| a__isNat(x(V1, V2)) | → | a__and(a__isNat(V1), isNat(V2)) | a__plus(N, 0) | → | a__U11(a__isNat(N), N) | |
| a__plus(N, s(M)) | → | a__U21(a__and(a__isNat(M), isNat(N)), M, N) | a__x(N, 0) | → | a__U31(a__isNat(N)) | |
| a__x(N, s(M)) | → | a__U41(a__and(a__isNat(M), isNat(N)), M, N) | mark(U11(X1, X2)) | → | a__U11(mark(X1), X2) | |
| mark(U21(X1, X2, X3)) | → | a__U21(mark(X1), X2, X3) | mark(plus(X1, X2)) | → | a__plus(mark(X1), mark(X2)) | |
| mark(U31(X)) | → | a__U31(mark(X)) | mark(U41(X1, X2, X3)) | → | a__U41(mark(X1), X2, X3) | |
| mark(x(X1, X2)) | → | a__x(mark(X1), mark(X2)) | mark(and(X1, X2)) | → | a__and(mark(X1), X2) | |
| mark(isNat(X)) | → | a__isNat(X) | mark(tt) | → | tt | |
| mark(s(X)) | → | s(mark(X)) | mark(0) | → | 0 | |
| a__U11(X1, X2) | → | U11(X1, X2) | a__U21(X1, X2, X3) | → | U21(X1, X2, X3) | |
| a__plus(X1, X2) | → | plus(X1, X2) | a__U31(X) | → | U31(X) | |
| a__U41(X1, X2, X3) | → | U41(X1, X2, X3) | a__x(X1, X2) | → | x(X1, X2) | |
| a__and(X1, X2) | → | and(X1, X2) | a__isNat(X) | → | isNat(X) |
Termination of terms over the following signature is verified: plus, a__plus, mark, and, a__and, isNat, 0, s, U41, tt, a__isNat, a__U41, U11, a__U31, U31, a__U11, a__x, U21, a__U21, x
| mark#(U11(X1, X2)) | → | mark#(X1) | a__isNat#(plus(V1, V2)) | → | a__and#(a__isNat(V1), isNat(V2)) | |
| mark#(isNat(X)) | → | a__isNat#(X) | a__isNat#(x(V1, V2)) | → | a__and#(a__isNat(V1), isNat(V2)) | |
| mark#(x(X1, X2)) | → | a__x#(mark(X1), mark(X2)) | mark#(U31(X)) | → | a__U31#(mark(X)) | |
| a__x#(N, s(M)) | → | a__and#(a__isNat(M), isNat(N)) | mark#(U41(X1, X2, X3)) | → | a__U41#(mark(X1), X2, X3) | |
| a__isNat#(s(V1)) | → | a__isNat#(V1) | a__plus#(N, 0) | → | a__U11#(a__isNat(N), N) | |
| a__U41#(tt, M, N) | → | mark#(M) | mark#(plus(X1, X2)) | → | mark#(X2) | |
| a__and#(tt, X) | → | mark#(X) | a__isNat#(plus(V1, V2)) | → | a__isNat#(V1) | |
| a__U41#(tt, M, N) | → | a__plus#(a__x(mark(N), mark(M)), mark(N)) | a__plus#(N, s(M)) | → | a__isNat#(M) | |
| mark#(s(X)) | → | mark#(X) | a__x#(N, 0) | → | a__isNat#(N) | |
| a__plus#(N, s(M)) | → | a__and#(a__isNat(M), isNat(N)) | a__U21#(tt, M, N) | → | mark#(M) | |
| a__U21#(tt, M, N) | → | mark#(N) | a__U11#(tt, N) | → | mark#(N) | |
| mark#(plus(X1, X2)) | → | a__plus#(mark(X1), mark(X2)) | a__U41#(tt, M, N) | → | a__x#(mark(N), mark(M)) | |
| a__x#(N, s(M)) | → | a__U41#(a__and(a__isNat(M), isNat(N)), M, N) | a__U21#(tt, M, N) | → | a__plus#(mark(N), mark(M)) | |
| mark#(U21(X1, X2, X3)) | → | a__U21#(mark(X1), X2, X3) | mark#(plus(X1, X2)) | → | mark#(X1) | |
| a__plus#(N, s(M)) | → | a__U21#(a__and(a__isNat(M), isNat(N)), M, N) | mark#(and(X1, X2)) | → | mark#(X1) | |
| mark#(x(X1, X2)) | → | mark#(X2) | a__x#(N, s(M)) | → | a__isNat#(M) | |
| mark#(U41(X1, X2, X3)) | → | mark#(X1) | mark#(and(X1, X2)) | → | a__and#(mark(X1), X2) | |
| a__plus#(N, 0) | → | a__isNat#(N) | a__U41#(tt, M, N) | → | mark#(N) | |
| mark#(U11(X1, X2)) | → | a__U11#(mark(X1), X2) | mark#(U31(X)) | → | mark#(X) | |
| mark#(U21(X1, X2, X3)) | → | mark#(X1) | a__x#(N, 0) | → | a__U31#(a__isNat(N)) | |
| a__isNat#(x(V1, V2)) | → | a__isNat#(V1) | mark#(x(X1, X2)) | → | mark#(X1) |
| a__U11(tt, N) | → | mark(N) | a__U21(tt, M, N) | → | s(a__plus(mark(N), mark(M))) | |
| a__U31(tt) | → | 0 | a__U41(tt, M, N) | → | a__plus(a__x(mark(N), mark(M)), mark(N)) | |
| a__and(tt, X) | → | mark(X) | a__isNat(0) | → | tt | |
| a__isNat(plus(V1, V2)) | → | a__and(a__isNat(V1), isNat(V2)) | a__isNat(s(V1)) | → | a__isNat(V1) | |
| a__isNat(x(V1, V2)) | → | a__and(a__isNat(V1), isNat(V2)) | a__plus(N, 0) | → | a__U11(a__isNat(N), N) | |
| a__plus(N, s(M)) | → | a__U21(a__and(a__isNat(M), isNat(N)), M, N) | a__x(N, 0) | → | a__U31(a__isNat(N)) | |
| a__x(N, s(M)) | → | a__U41(a__and(a__isNat(M), isNat(N)), M, N) | mark(U11(X1, X2)) | → | a__U11(mark(X1), X2) | |
| mark(U21(X1, X2, X3)) | → | a__U21(mark(X1), X2, X3) | mark(plus(X1, X2)) | → | a__plus(mark(X1), mark(X2)) | |
| mark(U31(X)) | → | a__U31(mark(X)) | mark(U41(X1, X2, X3)) | → | a__U41(mark(X1), X2, X3) | |
| mark(x(X1, X2)) | → | a__x(mark(X1), mark(X2)) | mark(and(X1, X2)) | → | a__and(mark(X1), X2) | |
| mark(isNat(X)) | → | a__isNat(X) | mark(tt) | → | tt | |
| mark(s(X)) | → | s(mark(X)) | mark(0) | → | 0 | |
| a__U11(X1, X2) | → | U11(X1, X2) | a__U21(X1, X2, X3) | → | U21(X1, X2, X3) | |
| a__plus(X1, X2) | → | plus(X1, X2) | a__U31(X) | → | U31(X) | |
| a__U41(X1, X2, X3) | → | U41(X1, X2, X3) | a__x(X1, X2) | → | x(X1, X2) | |
| a__and(X1, X2) | → | and(X1, X2) | a__isNat(X) | → | isNat(X) |
Termination of terms over the following signature is verified: plus, a__plus, mark, and, a__and, isNat, 0, s, U41, tt, a__isNat, a__U41, a__U31, U11, a__U11, U31, a__x, a__U21, U21, x
| mark#(U11(X1, X2)) → mark#(X1) | mark#(isNat(X)) → a__isNat#(X) |
| a__isNat#(plus(V1, V2)) → a__and#(a__isNat(V1), isNat(V2)) | a__isNat#(x(V1, V2)) → a__and#(a__isNat(V1), isNat(V2)) |
| mark#(x(X1, X2)) → a__x#(mark(X1), mark(X2)) | mark#(U41(X1, X2, X3)) → a__U41#(mark(X1), X2, X3) |
| a__x#(N, s(M)) → a__and#(a__isNat(M), isNat(N)) | a__isNat#(s(V1)) → a__isNat#(V1) |
| a__plus#(N, 0) → a__U11#(a__isNat(N), N) | a__U41#(tt, M, N) → mark#(M) |
| mark#(plus(X1, X2)) → mark#(X2) | a__and#(tt, X) → mark#(X) |
| a__isNat#(plus(V1, V2)) → a__isNat#(V1) | a__U41#(tt, M, N) → a__plus#(a__x(mark(N), mark(M)), mark(N)) |
| a__plus#(N, s(M)) → a__isNat#(M) | a__x#(N, 0) → a__isNat#(N) |
| mark#(s(X)) → mark#(X) | a__plus#(N, s(M)) → a__and#(a__isNat(M), isNat(N)) |
| a__U21#(tt, M, N) → mark#(M) | a__U21#(tt, M, N) → mark#(N) |
| a__U11#(tt, N) → mark#(N) | a__U41#(tt, M, N) → a__x#(mark(N), mark(M)) |
| mark#(plus(X1, X2)) → a__plus#(mark(X1), mark(X2)) | a__x#(N, s(M)) → a__U41#(a__and(a__isNat(M), isNat(N)), M, N) |
| mark#(U21(X1, X2, X3)) → a__U21#(mark(X1), X2, X3) | a__U21#(tt, M, N) → a__plus#(mark(N), mark(M)) |
| mark#(plus(X1, X2)) → mark#(X1) | a__plus#(N, s(M)) → a__U21#(a__and(a__isNat(M), isNat(N)), M, N) |
| mark#(and(X1, X2)) → mark#(X1) | mark#(x(X1, X2)) → mark#(X2) |
| a__x#(N, s(M)) → a__isNat#(M) | mark#(U41(X1, X2, X3)) → mark#(X1) |
| mark#(and(X1, X2)) → a__and#(mark(X1), X2) | a__plus#(N, 0) → a__isNat#(N) |
| a__U41#(tt, M, N) → mark#(N) | mark#(U11(X1, X2)) → a__U11#(mark(X1), X2) |
| mark#(U31(X)) → mark#(X) | mark#(U21(X1, X2, X3)) → mark#(X1) |
| a__isNat#(x(V1, V2)) → a__isNat#(V1) | mark#(x(X1, X2)) → mark#(X1) |