TIMEOUT
The TRS could not be proven terminating. The proof attempt took 60023 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (5413ms).
| – Problem 2 was processed with processor SubtermCriterion (2ms).
| | – Problem 10 was processed with processor ReductionPairSAT (440ms).
| | | – Problem 17 was processed with processor ReductionPairSAT (289ms).
| | | | – Problem 23 remains open; application of the following processors failed [].
| – Problem 3 was processed with processor SubtermCriterion (2ms).
| | – Problem 11 was processed with processor ReductionPairSAT (450ms).
| | | – Problem 18 remains open; application of the following processors failed [DependencyGraph (7ms)].
| – Problem 4 was processed with processor SubtermCriterion (1ms).
| | – Problem 12 was processed with processor ReductionPairSAT (72ms).
| | | – Problem 19 remains open; application of the following processors failed [DependencyGraph (1ms)].
| – Problem 5 was processed with processor SubtermCriterion (1ms).
| | – Problem 13 was processed with processor ReductionPairSAT (409ms).
| | | – Problem 20 remains open; application of the following processors failed [DependencyGraph (7ms)].
| – Problem 6 was processed with processor ReductionPairSAT (19530ms).
| | – Problem 16 remains open; application of the following processors failed [DependencyGraph (515ms), ReductionPairSAT (timeout)].
| – Problem 7 was processed with processor SubtermCriterion (3ms).
| | – Problem 14 was processed with processor ReductionPairSAT (161ms).
| | | – Problem 21 remains open; application of the following processors failed [DependencyGraph (2ms)].
| – Problem 8 was processed with processor SubtermCriterion (1ms).
| – Problem 9 was processed with processor SubtermCriterion (2ms).
| | – Problem 15 was processed with processor ReductionPairSAT (410ms).
| | | – Problem 22 remains open; application of the following processors failed [DependencyGraph (6ms)].
The following open problems remain:
Open Dependency Pair Problem 17
Dependency Pairs
| U21#(X1, active(X2), X3) | → | U21#(X1, X2, X3) | | U21#(X1, X2, mark(X3)) | → | U21#(X1, X2, X3) |
| U21#(X1, X2, active(X3)) | → | U21#(X1, X2, X3) |
Rewrite Rules
| active(U11(tt, M, N)) | → | mark(U12(tt, M, N)) | | active(U12(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U21(tt, M, N)) | → | mark(U22(tt, M, N)) | | active(U22(tt, M, N)) | → | mark(plus(x(N, M), N)) |
| active(plus(N, 0)) | → | mark(N) | | active(plus(N, s(M))) | → | mark(U11(tt, M, N)) |
| active(x(N, 0)) | → | mark(0) | | active(x(N, s(M))) | → | mark(U21(tt, M, N)) |
| mark(U11(X1, X2, X3)) | → | active(U11(mark(X1), X2, X3)) | | mark(tt) | → | active(tt) |
| mark(U12(X1, X2, X3)) | → | active(U12(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U21(X1, X2, X3)) | → | active(U21(mark(X1), X2, X3)) |
| mark(U22(X1, X2, X3)) | → | active(U22(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| mark(0) | → | active(0) | | U11(mark(X1), X2, X3) | → | U11(X1, X2, X3) |
| U11(X1, mark(X2), X3) | → | U11(X1, X2, X3) | | U11(X1, X2, mark(X3)) | → | U11(X1, X2, X3) |
| U11(active(X1), X2, X3) | → | U11(X1, X2, X3) | | U11(X1, active(X2), X3) | → | U11(X1, X2, X3) |
| U11(X1, X2, active(X3)) | → | U11(X1, X2, X3) | | U12(mark(X1), X2, X3) | → | U12(X1, X2, X3) |
| U12(X1, mark(X2), X3) | → | U12(X1, X2, X3) | | U12(X1, X2, mark(X3)) | → | U12(X1, X2, X3) |
| U12(active(X1), X2, X3) | → | U12(X1, X2, X3) | | U12(X1, active(X2), X3) | → | U12(X1, X2, X3) |
| U12(X1, X2, active(X3)) | → | U12(X1, X2, X3) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | plus(mark(X1), X2) | → | plus(X1, X2) |
| plus(X1, mark(X2)) | → | plus(X1, X2) | | plus(active(X1), X2) | → | plus(X1, X2) |
| plus(X1, active(X2)) | → | plus(X1, X2) | | U21(mark(X1), X2, X3) | → | U21(X1, X2, X3) |
| U21(X1, mark(X2), X3) | → | U21(X1, X2, X3) | | U21(X1, X2, mark(X3)) | → | U21(X1, X2, X3) |
| U21(active(X1), X2, X3) | → | U21(X1, X2, X3) | | U21(X1, active(X2), X3) | → | U21(X1, X2, X3) |
| U21(X1, X2, active(X3)) | → | U21(X1, X2, X3) | | U22(mark(X1), X2, X3) | → | U22(X1, X2, X3) |
| U22(X1, mark(X2), X3) | → | U22(X1, X2, X3) | | U22(X1, X2, mark(X3)) | → | U22(X1, X2, X3) |
| U22(active(X1), X2, X3) | → | U22(X1, X2, X3) | | U22(X1, active(X2), X3) | → | U22(X1, X2, X3) |
| U22(X1, X2, active(X3)) | → | U22(X1, X2, X3) | | x(mark(X1), X2) | → | x(X1, X2) |
| x(X1, mark(X2)) | → | x(X1, X2) | | x(active(X1), X2) | → | x(X1, X2) |
| x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, tt, U11, active, U12, mark, U21, x, U22
Open Dependency Pair Problem 16
Dependency Pairs
| active#(U11(tt, M, N)) | → | mark#(U12(tt, M, N)) | | active#(U22(tt, M, N)) | → | mark#(plus(x(N, M), N)) |
| active#(U21(tt, M, N)) | → | mark#(U22(tt, M, N)) | | mark#(tt) | → | active#(tt) |
| active#(plus(N, s(M))) | → | mark#(U11(tt, M, N)) | | active#(U12(tt, M, N)) | → | mark#(s(plus(N, M))) |
| mark#(U11(X1, X2, X3)) | → | active#(U11(mark(X1), X2, X3)) | | active#(x(N, s(M))) | → | mark#(U21(tt, M, N)) |
| mark#(plus(X1, X2)) | → | mark#(X2) | | mark#(U12(X1, X2, X3)) | → | active#(U12(mark(X1), X2, X3)) |
| mark#(s(X)) | → | mark#(X) | | mark#(U12(X1, X2, X3)) | → | mark#(X1) |
| mark#(plus(X1, X2)) | → | active#(plus(mark(X1), mark(X2))) | | mark#(s(X)) | → | active#(s(mark(X))) |
| mark#(x(X1, X2)) | → | active#(x(mark(X1), mark(X2))) | | active#(plus(N, 0)) | → | mark#(N) |
| mark#(U22(X1, X2, X3)) | → | mark#(X1) | | mark#(U22(X1, X2, X3)) | → | active#(U22(mark(X1), X2, X3)) |
| mark#(plus(X1, X2)) | → | mark#(X1) | | active#(x(N, 0)) | → | mark#(0) |
| mark#(x(X1, X2)) | → | mark#(X2) | | mark#(U21(X1, X2, X3)) | → | mark#(X1) |
| mark#(U21(X1, X2, X3)) | → | active#(U21(mark(X1), X2, X3)) | | mark#(U11(X1, X2, X3)) | → | mark#(X1) |
| mark#(x(X1, X2)) | → | mark#(X1) |
Rewrite Rules
| active(U11(tt, M, N)) | → | mark(U12(tt, M, N)) | | active(U12(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U21(tt, M, N)) | → | mark(U22(tt, M, N)) | | active(U22(tt, M, N)) | → | mark(plus(x(N, M), N)) |
| active(plus(N, 0)) | → | mark(N) | | active(plus(N, s(M))) | → | mark(U11(tt, M, N)) |
| active(x(N, 0)) | → | mark(0) | | active(x(N, s(M))) | → | mark(U21(tt, M, N)) |
| mark(U11(X1, X2, X3)) | → | active(U11(mark(X1), X2, X3)) | | mark(tt) | → | active(tt) |
| mark(U12(X1, X2, X3)) | → | active(U12(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U21(X1, X2, X3)) | → | active(U21(mark(X1), X2, X3)) |
| mark(U22(X1, X2, X3)) | → | active(U22(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| mark(0) | → | active(0) | | U11(mark(X1), X2, X3) | → | U11(X1, X2, X3) |
| U11(X1, mark(X2), X3) | → | U11(X1, X2, X3) | | U11(X1, X2, mark(X3)) | → | U11(X1, X2, X3) |
| U11(active(X1), X2, X3) | → | U11(X1, X2, X3) | | U11(X1, active(X2), X3) | → | U11(X1, X2, X3) |
| U11(X1, X2, active(X3)) | → | U11(X1, X2, X3) | | U12(mark(X1), X2, X3) | → | U12(X1, X2, X3) |
| U12(X1, mark(X2), X3) | → | U12(X1, X2, X3) | | U12(X1, X2, mark(X3)) | → | U12(X1, X2, X3) |
| U12(active(X1), X2, X3) | → | U12(X1, X2, X3) | | U12(X1, active(X2), X3) | → | U12(X1, X2, X3) |
| U12(X1, X2, active(X3)) | → | U12(X1, X2, X3) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | plus(mark(X1), X2) | → | plus(X1, X2) |
| plus(X1, mark(X2)) | → | plus(X1, X2) | | plus(active(X1), X2) | → | plus(X1, X2) |
| plus(X1, active(X2)) | → | plus(X1, X2) | | U21(mark(X1), X2, X3) | → | U21(X1, X2, X3) |
| U21(X1, mark(X2), X3) | → | U21(X1, X2, X3) | | U21(X1, X2, mark(X3)) | → | U21(X1, X2, X3) |
| U21(active(X1), X2, X3) | → | U21(X1, X2, X3) | | U21(X1, active(X2), X3) | → | U21(X1, X2, X3) |
| U21(X1, X2, active(X3)) | → | U21(X1, X2, X3) | | U22(mark(X1), X2, X3) | → | U22(X1, X2, X3) |
| U22(X1, mark(X2), X3) | → | U22(X1, X2, X3) | | U22(X1, X2, mark(X3)) | → | U22(X1, X2, X3) |
| U22(active(X1), X2, X3) | → | U22(X1, X2, X3) | | U22(X1, active(X2), X3) | → | U22(X1, X2, X3) |
| U22(X1, X2, active(X3)) | → | U22(X1, X2, X3) | | x(mark(X1), X2) | → | x(X1, X2) |
| x(X1, mark(X2)) | → | x(X1, X2) | | x(active(X1), X2) | → | x(X1, X2) |
| x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, tt, active, U11, mark, U12, U21, U22, x
Open Dependency Pair Problem 19
Dependency Pairs
| x#(X1, mark(X2)) | → | x#(X1, X2) |
Rewrite Rules
| active(U11(tt, M, N)) | → | mark(U12(tt, M, N)) | | active(U12(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U21(tt, M, N)) | → | mark(U22(tt, M, N)) | | active(U22(tt, M, N)) | → | mark(plus(x(N, M), N)) |
| active(plus(N, 0)) | → | mark(N) | | active(plus(N, s(M))) | → | mark(U11(tt, M, N)) |
| active(x(N, 0)) | → | mark(0) | | active(x(N, s(M))) | → | mark(U21(tt, M, N)) |
| mark(U11(X1, X2, X3)) | → | active(U11(mark(X1), X2, X3)) | | mark(tt) | → | active(tt) |
| mark(U12(X1, X2, X3)) | → | active(U12(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U21(X1, X2, X3)) | → | active(U21(mark(X1), X2, X3)) |
| mark(U22(X1, X2, X3)) | → | active(U22(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| mark(0) | → | active(0) | | U11(mark(X1), X2, X3) | → | U11(X1, X2, X3) |
| U11(X1, mark(X2), X3) | → | U11(X1, X2, X3) | | U11(X1, X2, mark(X3)) | → | U11(X1, X2, X3) |
| U11(active(X1), X2, X3) | → | U11(X1, X2, X3) | | U11(X1, active(X2), X3) | → | U11(X1, X2, X3) |
| U11(X1, X2, active(X3)) | → | U11(X1, X2, X3) | | U12(mark(X1), X2, X3) | → | U12(X1, X2, X3) |
| U12(X1, mark(X2), X3) | → | U12(X1, X2, X3) | | U12(X1, X2, mark(X3)) | → | U12(X1, X2, X3) |
| U12(active(X1), X2, X3) | → | U12(X1, X2, X3) | | U12(X1, active(X2), X3) | → | U12(X1, X2, X3) |
| U12(X1, X2, active(X3)) | → | U12(X1, X2, X3) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | plus(mark(X1), X2) | → | plus(X1, X2) |
| plus(X1, mark(X2)) | → | plus(X1, X2) | | plus(active(X1), X2) | → | plus(X1, X2) |
| plus(X1, active(X2)) | → | plus(X1, X2) | | U21(mark(X1), X2, X3) | → | U21(X1, X2, X3) |
| U21(X1, mark(X2), X3) | → | U21(X1, X2, X3) | | U21(X1, X2, mark(X3)) | → | U21(X1, X2, X3) |
| U21(active(X1), X2, X3) | → | U21(X1, X2, X3) | | U21(X1, active(X2), X3) | → | U21(X1, X2, X3) |
| U21(X1, X2, active(X3)) | → | U21(X1, X2, X3) | | U22(mark(X1), X2, X3) | → | U22(X1, X2, X3) |
| U22(X1, mark(X2), X3) | → | U22(X1, X2, X3) | | U22(X1, X2, mark(X3)) | → | U22(X1, X2, X3) |
| U22(active(X1), X2, X3) | → | U22(X1, X2, X3) | | U22(X1, active(X2), X3) | → | U22(X1, X2, X3) |
| U22(X1, X2, active(X3)) | → | U22(X1, X2, X3) | | x(mark(X1), X2) | → | x(X1, X2) |
| x(X1, mark(X2)) | → | x(X1, X2) | | x(active(X1), X2) | → | x(X1, X2) |
| x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, tt, U11, active, U12, mark, U21, x, U22
Open Dependency Pair Problem 18
Dependency Pairs
| U22#(X1, active(X2), X3) | → | U22#(X1, X2, X3) | | U22#(X1, X2, mark(X3)) | → | U22#(X1, X2, X3) |
| U22#(X1, mark(X2), X3) | → | U22#(X1, X2, X3) |
Rewrite Rules
| active(U11(tt, M, N)) | → | mark(U12(tt, M, N)) | | active(U12(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U21(tt, M, N)) | → | mark(U22(tt, M, N)) | | active(U22(tt, M, N)) | → | mark(plus(x(N, M), N)) |
| active(plus(N, 0)) | → | mark(N) | | active(plus(N, s(M))) | → | mark(U11(tt, M, N)) |
| active(x(N, 0)) | → | mark(0) | | active(x(N, s(M))) | → | mark(U21(tt, M, N)) |
| mark(U11(X1, X2, X3)) | → | active(U11(mark(X1), X2, X3)) | | mark(tt) | → | active(tt) |
| mark(U12(X1, X2, X3)) | → | active(U12(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U21(X1, X2, X3)) | → | active(U21(mark(X1), X2, X3)) |
| mark(U22(X1, X2, X3)) | → | active(U22(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| mark(0) | → | active(0) | | U11(mark(X1), X2, X3) | → | U11(X1, X2, X3) |
| U11(X1, mark(X2), X3) | → | U11(X1, X2, X3) | | U11(X1, X2, mark(X3)) | → | U11(X1, X2, X3) |
| U11(active(X1), X2, X3) | → | U11(X1, X2, X3) | | U11(X1, active(X2), X3) | → | U11(X1, X2, X3) |
| U11(X1, X2, active(X3)) | → | U11(X1, X2, X3) | | U12(mark(X1), X2, X3) | → | U12(X1, X2, X3) |
| U12(X1, mark(X2), X3) | → | U12(X1, X2, X3) | | U12(X1, X2, mark(X3)) | → | U12(X1, X2, X3) |
| U12(active(X1), X2, X3) | → | U12(X1, X2, X3) | | U12(X1, active(X2), X3) | → | U12(X1, X2, X3) |
| U12(X1, X2, active(X3)) | → | U12(X1, X2, X3) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | plus(mark(X1), X2) | → | plus(X1, X2) |
| plus(X1, mark(X2)) | → | plus(X1, X2) | | plus(active(X1), X2) | → | plus(X1, X2) |
| plus(X1, active(X2)) | → | plus(X1, X2) | | U21(mark(X1), X2, X3) | → | U21(X1, X2, X3) |
| U21(X1, mark(X2), X3) | → | U21(X1, X2, X3) | | U21(X1, X2, mark(X3)) | → | U21(X1, X2, X3) |
| U21(active(X1), X2, X3) | → | U21(X1, X2, X3) | | U21(X1, active(X2), X3) | → | U21(X1, X2, X3) |
| U21(X1, X2, active(X3)) | → | U21(X1, X2, X3) | | U22(mark(X1), X2, X3) | → | U22(X1, X2, X3) |
| U22(X1, mark(X2), X3) | → | U22(X1, X2, X3) | | U22(X1, X2, mark(X3)) | → | U22(X1, X2, X3) |
| U22(active(X1), X2, X3) | → | U22(X1, X2, X3) | | U22(X1, active(X2), X3) | → | U22(X1, X2, X3) |
| U22(X1, X2, active(X3)) | → | U22(X1, X2, X3) | | x(mark(X1), X2) | → | x(X1, X2) |
| x(X1, mark(X2)) | → | x(X1, X2) | | x(active(X1), X2) | → | x(X1, X2) |
| x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, tt, U11, active, U12, mark, U21, x, U22
Open Dependency Pair Problem 21
Dependency Pairs
| plus#(X1, mark(X2)) | → | plus#(X1, X2) |
Rewrite Rules
| active(U11(tt, M, N)) | → | mark(U12(tt, M, N)) | | active(U12(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U21(tt, M, N)) | → | mark(U22(tt, M, N)) | | active(U22(tt, M, N)) | → | mark(plus(x(N, M), N)) |
| active(plus(N, 0)) | → | mark(N) | | active(plus(N, s(M))) | → | mark(U11(tt, M, N)) |
| active(x(N, 0)) | → | mark(0) | | active(x(N, s(M))) | → | mark(U21(tt, M, N)) |
| mark(U11(X1, X2, X3)) | → | active(U11(mark(X1), X2, X3)) | | mark(tt) | → | active(tt) |
| mark(U12(X1, X2, X3)) | → | active(U12(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U21(X1, X2, X3)) | → | active(U21(mark(X1), X2, X3)) |
| mark(U22(X1, X2, X3)) | → | active(U22(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| mark(0) | → | active(0) | | U11(mark(X1), X2, X3) | → | U11(X1, X2, X3) |
| U11(X1, mark(X2), X3) | → | U11(X1, X2, X3) | | U11(X1, X2, mark(X3)) | → | U11(X1, X2, X3) |
| U11(active(X1), X2, X3) | → | U11(X1, X2, X3) | | U11(X1, active(X2), X3) | → | U11(X1, X2, X3) |
| U11(X1, X2, active(X3)) | → | U11(X1, X2, X3) | | U12(mark(X1), X2, X3) | → | U12(X1, X2, X3) |
| U12(X1, mark(X2), X3) | → | U12(X1, X2, X3) | | U12(X1, X2, mark(X3)) | → | U12(X1, X2, X3) |
| U12(active(X1), X2, X3) | → | U12(X1, X2, X3) | | U12(X1, active(X2), X3) | → | U12(X1, X2, X3) |
| U12(X1, X2, active(X3)) | → | U12(X1, X2, X3) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | plus(mark(X1), X2) | → | plus(X1, X2) |
| plus(X1, mark(X2)) | → | plus(X1, X2) | | plus(active(X1), X2) | → | plus(X1, X2) |
| plus(X1, active(X2)) | → | plus(X1, X2) | | U21(mark(X1), X2, X3) | → | U21(X1, X2, X3) |
| U21(X1, mark(X2), X3) | → | U21(X1, X2, X3) | | U21(X1, X2, mark(X3)) | → | U21(X1, X2, X3) |
| U21(active(X1), X2, X3) | → | U21(X1, X2, X3) | | U21(X1, active(X2), X3) | → | U21(X1, X2, X3) |
| U21(X1, X2, active(X3)) | → | U21(X1, X2, X3) | | U22(mark(X1), X2, X3) | → | U22(X1, X2, X3) |
| U22(X1, mark(X2), X3) | → | U22(X1, X2, X3) | | U22(X1, X2, mark(X3)) | → | U22(X1, X2, X3) |
| U22(active(X1), X2, X3) | → | U22(X1, X2, X3) | | U22(X1, active(X2), X3) | → | U22(X1, X2, X3) |
| U22(X1, X2, active(X3)) | → | U22(X1, X2, X3) | | x(mark(X1), X2) | → | x(X1, X2) |
| x(X1, mark(X2)) | → | x(X1, X2) | | x(active(X1), X2) | → | x(X1, X2) |
| x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, tt, U11, active, U12, mark, U21, x, U22
Open Dependency Pair Problem 20
Dependency Pairs
| U12#(X1, active(X2), X3) | → | U12#(X1, X2, X3) | | U12#(X1, X2, mark(X3)) | → | U12#(X1, X2, X3) |
| U12#(X1, mark(X2), X3) | → | U12#(X1, X2, X3) |
Rewrite Rules
| active(U11(tt, M, N)) | → | mark(U12(tt, M, N)) | | active(U12(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U21(tt, M, N)) | → | mark(U22(tt, M, N)) | | active(U22(tt, M, N)) | → | mark(plus(x(N, M), N)) |
| active(plus(N, 0)) | → | mark(N) | | active(plus(N, s(M))) | → | mark(U11(tt, M, N)) |
| active(x(N, 0)) | → | mark(0) | | active(x(N, s(M))) | → | mark(U21(tt, M, N)) |
| mark(U11(X1, X2, X3)) | → | active(U11(mark(X1), X2, X3)) | | mark(tt) | → | active(tt) |
| mark(U12(X1, X2, X3)) | → | active(U12(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U21(X1, X2, X3)) | → | active(U21(mark(X1), X2, X3)) |
| mark(U22(X1, X2, X3)) | → | active(U22(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| mark(0) | → | active(0) | | U11(mark(X1), X2, X3) | → | U11(X1, X2, X3) |
| U11(X1, mark(X2), X3) | → | U11(X1, X2, X3) | | U11(X1, X2, mark(X3)) | → | U11(X1, X2, X3) |
| U11(active(X1), X2, X3) | → | U11(X1, X2, X3) | | U11(X1, active(X2), X3) | → | U11(X1, X2, X3) |
| U11(X1, X2, active(X3)) | → | U11(X1, X2, X3) | | U12(mark(X1), X2, X3) | → | U12(X1, X2, X3) |
| U12(X1, mark(X2), X3) | → | U12(X1, X2, X3) | | U12(X1, X2, mark(X3)) | → | U12(X1, X2, X3) |
| U12(active(X1), X2, X3) | → | U12(X1, X2, X3) | | U12(X1, active(X2), X3) | → | U12(X1, X2, X3) |
| U12(X1, X2, active(X3)) | → | U12(X1, X2, X3) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | plus(mark(X1), X2) | → | plus(X1, X2) |
| plus(X1, mark(X2)) | → | plus(X1, X2) | | plus(active(X1), X2) | → | plus(X1, X2) |
| plus(X1, active(X2)) | → | plus(X1, X2) | | U21(mark(X1), X2, X3) | → | U21(X1, X2, X3) |
| U21(X1, mark(X2), X3) | → | U21(X1, X2, X3) | | U21(X1, X2, mark(X3)) | → | U21(X1, X2, X3) |
| U21(active(X1), X2, X3) | → | U21(X1, X2, X3) | | U21(X1, active(X2), X3) | → | U21(X1, X2, X3) |
| U21(X1, X2, active(X3)) | → | U21(X1, X2, X3) | | U22(mark(X1), X2, X3) | → | U22(X1, X2, X3) |
| U22(X1, mark(X2), X3) | → | U22(X1, X2, X3) | | U22(X1, X2, mark(X3)) | → | U22(X1, X2, X3) |
| U22(active(X1), X2, X3) | → | U22(X1, X2, X3) | | U22(X1, active(X2), X3) | → | U22(X1, X2, X3) |
| U22(X1, X2, active(X3)) | → | U22(X1, X2, X3) | | x(mark(X1), X2) | → | x(X1, X2) |
| x(X1, mark(X2)) | → | x(X1, X2) | | x(active(X1), X2) | → | x(X1, X2) |
| x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, tt, U11, active, U12, mark, U21, x, U22
Open Dependency Pair Problem 22
Dependency Pairs
| U11#(X1, X2, mark(X3)) | → | U11#(X1, X2, X3) | | U11#(X1, mark(X2), X3) | → | U11#(X1, X2, X3) |
| U11#(X1, X2, active(X3)) | → | U11#(X1, X2, X3) |
Rewrite Rules
| active(U11(tt, M, N)) | → | mark(U12(tt, M, N)) | | active(U12(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U21(tt, M, N)) | → | mark(U22(tt, M, N)) | | active(U22(tt, M, N)) | → | mark(plus(x(N, M), N)) |
| active(plus(N, 0)) | → | mark(N) | | active(plus(N, s(M))) | → | mark(U11(tt, M, N)) |
| active(x(N, 0)) | → | mark(0) | | active(x(N, s(M))) | → | mark(U21(tt, M, N)) |
| mark(U11(X1, X2, X3)) | → | active(U11(mark(X1), X2, X3)) | | mark(tt) | → | active(tt) |
| mark(U12(X1, X2, X3)) | → | active(U12(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U21(X1, X2, X3)) | → | active(U21(mark(X1), X2, X3)) |
| mark(U22(X1, X2, X3)) | → | active(U22(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| mark(0) | → | active(0) | | U11(mark(X1), X2, X3) | → | U11(X1, X2, X3) |
| U11(X1, mark(X2), X3) | → | U11(X1, X2, X3) | | U11(X1, X2, mark(X3)) | → | U11(X1, X2, X3) |
| U11(active(X1), X2, X3) | → | U11(X1, X2, X3) | | U11(X1, active(X2), X3) | → | U11(X1, X2, X3) |
| U11(X1, X2, active(X3)) | → | U11(X1, X2, X3) | | U12(mark(X1), X2, X3) | → | U12(X1, X2, X3) |
| U12(X1, mark(X2), X3) | → | U12(X1, X2, X3) | | U12(X1, X2, mark(X3)) | → | U12(X1, X2, X3) |
| U12(active(X1), X2, X3) | → | U12(X1, X2, X3) | | U12(X1, active(X2), X3) | → | U12(X1, X2, X3) |
| U12(X1, X2, active(X3)) | → | U12(X1, X2, X3) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | plus(mark(X1), X2) | → | plus(X1, X2) |
| plus(X1, mark(X2)) | → | plus(X1, X2) | | plus(active(X1), X2) | → | plus(X1, X2) |
| plus(X1, active(X2)) | → | plus(X1, X2) | | U21(mark(X1), X2, X3) | → | U21(X1, X2, X3) |
| U21(X1, mark(X2), X3) | → | U21(X1, X2, X3) | | U21(X1, X2, mark(X3)) | → | U21(X1, X2, X3) |
| U21(active(X1), X2, X3) | → | U21(X1, X2, X3) | | U21(X1, active(X2), X3) | → | U21(X1, X2, X3) |
| U21(X1, X2, active(X3)) | → | U21(X1, X2, X3) | | U22(mark(X1), X2, X3) | → | U22(X1, X2, X3) |
| U22(X1, mark(X2), X3) | → | U22(X1, X2, X3) | | U22(X1, X2, mark(X3)) | → | U22(X1, X2, X3) |
| U22(active(X1), X2, X3) | → | U22(X1, X2, X3) | | U22(X1, active(X2), X3) | → | U22(X1, X2, X3) |
| U22(X1, X2, active(X3)) | → | U22(X1, X2, X3) | | x(mark(X1), X2) | → | x(X1, X2) |
| x(X1, mark(X2)) | → | x(X1, X2) | | x(active(X1), X2) | → | x(X1, X2) |
| x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, tt, U11, active, U12, mark, U21, x, U22
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| U22#(X1, active(X2), X3) | → | U22#(X1, X2, X3) | | active#(x(N, s(M))) | → | U21#(tt, M, N) |
| active#(U11(tt, M, N)) | → | mark#(U12(tt, M, N)) | | active#(U21(tt, M, N)) | → | mark#(U22(tt, M, N)) |
| U22#(mark(X1), X2, X3) | → | U22#(X1, X2, X3) | | active#(plus(N, s(M))) | → | mark#(U11(tt, M, N)) |
| U21#(X1, X2, active(X3)) | → | U21#(X1, X2, X3) | | mark#(U11(X1, X2, X3)) | → | U11#(mark(X1), X2, X3) |
| x#(active(X1), X2) | → | x#(X1, X2) | | U22#(X1, X2, mark(X3)) | → | U22#(X1, X2, X3) |
| mark#(s(X)) | → | s#(mark(X)) | | mark#(U11(X1, X2, X3)) | → | active#(U11(mark(X1), X2, X3)) |
| x#(X1, active(X2)) | → | x#(X1, X2) | | plus#(X1, mark(X2)) | → | plus#(X1, X2) |
| mark#(U12(X1, X2, X3)) | → | active#(U12(mark(X1), X2, X3)) | | mark#(s(X)) | → | mark#(X) |
| x#(X1, mark(X2)) | → | x#(X1, X2) | | mark#(plus(X1, X2)) | → | active#(plus(mark(X1), mark(X2))) |
| U21#(X1, mark(X2), X3) | → | U21#(X1, X2, X3) | | U12#(mark(X1), X2, X3) | → | U12#(X1, X2, X3) |
| U11#(X1, active(X2), X3) | → | U11#(X1, X2, X3) | | active#(plus(N, 0)) | → | mark#(N) |
| U11#(mark(X1), X2, X3) | → | U11#(X1, X2, X3) | | U21#(mark(X1), X2, X3) | → | U21#(X1, X2, X3) |
| active#(x(N, 0)) | → | mark#(0) | | mark#(plus(X1, X2)) | → | mark#(X1) |
| mark#(x(X1, X2)) | → | mark#(X2) | | U12#(X1, active(X2), X3) | → | U12#(X1, X2, X3) |
| U21#(X1, active(X2), X3) | → | U21#(X1, X2, X3) | | U21#(active(X1), X2, X3) | → | U21#(X1, X2, X3) |
| U12#(X1, X2, mark(X3)) | → | U12#(X1, X2, X3) | | active#(U12(tt, M, N)) | → | s#(plus(N, M)) |
| plus#(X1, active(X2)) | → | plus#(X1, X2) | | mark#(U21(X1, X2, X3)) | → | mark#(X1) |
| plus#(mark(X1), X2) | → | plus#(X1, X2) | | mark#(U11(X1, X2, X3)) | → | mark#(X1) |
| plus#(active(X1), X2) | → | plus#(X1, X2) | | mark#(tt) | → | active#(tt) |
| active#(U22(tt, M, N)) | → | mark#(plus(x(N, M), N)) | | U12#(X1, mark(X2), X3) | → | U12#(X1, X2, X3) |
| active#(U12(tt, M, N)) | → | mark#(s(plus(N, M))) | | active#(x(N, s(M))) | → | mark#(U21(tt, M, N)) |
| x#(mark(X1), X2) | → | x#(X1, X2) | | U11#(X1, X2, mark(X3)) | → | U11#(X1, X2, X3) |
| mark#(plus(X1, X2)) | → | mark#(X2) | | U12#(active(X1), X2, X3) | → | U12#(X1, X2, X3) |
| mark#(U22(X1, X2, X3)) | → | U22#(mark(X1), X2, X3) | | U22#(active(X1), X2, X3) | → | U22#(X1, X2, X3) |
| U11#(active(X1), X2, X3) | → | U11#(X1, X2, X3) | | U21#(X1, X2, mark(X3)) | → | U21#(X1, X2, X3) |
| mark#(U12(X1, X2, X3)) | → | mark#(X1) | | U22#(X1, mark(X2), X3) | → | U22#(X1, X2, X3) |
| mark#(0) | → | active#(0) | | mark#(s(X)) | → | active#(s(mark(X))) |
| active#(U21(tt, M, N)) | → | U22#(tt, M, N) | | U12#(X1, X2, active(X3)) | → | U12#(X1, X2, X3) |
| mark#(x(X1, X2)) | → | active#(x(mark(X1), mark(X2))) | | active#(U22(tt, M, N)) | → | x#(N, M) |
| active#(U12(tt, M, N)) | → | plus#(N, M) | | mark#(U22(X1, X2, X3)) | → | mark#(X1) |
| mark#(plus(X1, X2)) | → | plus#(mark(X1), mark(X2)) | | mark#(U22(X1, X2, X3)) | → | active#(U22(mark(X1), X2, X3)) |
| mark#(U21(X1, X2, X3)) | → | U21#(mark(X1), X2, X3) | | mark#(U12(X1, X2, X3)) | → | U12#(mark(X1), X2, X3) |
| s#(mark(X)) | → | s#(X) | | active#(U22(tt, M, N)) | → | plus#(x(N, M), N) |
| active#(U11(tt, M, N)) | → | U12#(tt, M, N) | | U11#(X1, mark(X2), X3) | → | U11#(X1, X2, X3) |
| U22#(X1, X2, active(X3)) | → | U22#(X1, X2, X3) | | active#(plus(N, s(M))) | → | U11#(tt, M, N) |
| s#(active(X)) | → | s#(X) | | mark#(x(X1, X2)) | → | x#(mark(X1), mark(X2)) |
| mark#(U21(X1, X2, X3)) | → | active#(U21(mark(X1), X2, X3)) | | U11#(X1, X2, active(X3)) | → | U11#(X1, X2, X3) |
| mark#(x(X1, X2)) | → | mark#(X1) |
Rewrite Rules
| active(U11(tt, M, N)) | → | mark(U12(tt, M, N)) | | active(U12(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U21(tt, M, N)) | → | mark(U22(tt, M, N)) | | active(U22(tt, M, N)) | → | mark(plus(x(N, M), N)) |
| active(plus(N, 0)) | → | mark(N) | | active(plus(N, s(M))) | → | mark(U11(tt, M, N)) |
| active(x(N, 0)) | → | mark(0) | | active(x(N, s(M))) | → | mark(U21(tt, M, N)) |
| mark(U11(X1, X2, X3)) | → | active(U11(mark(X1), X2, X3)) | | mark(tt) | → | active(tt) |
| mark(U12(X1, X2, X3)) | → | active(U12(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U21(X1, X2, X3)) | → | active(U21(mark(X1), X2, X3)) |
| mark(U22(X1, X2, X3)) | → | active(U22(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| mark(0) | → | active(0) | | U11(mark(X1), X2, X3) | → | U11(X1, X2, X3) |
| U11(X1, mark(X2), X3) | → | U11(X1, X2, X3) | | U11(X1, X2, mark(X3)) | → | U11(X1, X2, X3) |
| U11(active(X1), X2, X3) | → | U11(X1, X2, X3) | | U11(X1, active(X2), X3) | → | U11(X1, X2, X3) |
| U11(X1, X2, active(X3)) | → | U11(X1, X2, X3) | | U12(mark(X1), X2, X3) | → | U12(X1, X2, X3) |
| U12(X1, mark(X2), X3) | → | U12(X1, X2, X3) | | U12(X1, X2, mark(X3)) | → | U12(X1, X2, X3) |
| U12(active(X1), X2, X3) | → | U12(X1, X2, X3) | | U12(X1, active(X2), X3) | → | U12(X1, X2, X3) |
| U12(X1, X2, active(X3)) | → | U12(X1, X2, X3) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | plus(mark(X1), X2) | → | plus(X1, X2) |
| plus(X1, mark(X2)) | → | plus(X1, X2) | | plus(active(X1), X2) | → | plus(X1, X2) |
| plus(X1, active(X2)) | → | plus(X1, X2) | | U21(mark(X1), X2, X3) | → | U21(X1, X2, X3) |
| U21(X1, mark(X2), X3) | → | U21(X1, X2, X3) | | U21(X1, X2, mark(X3)) | → | U21(X1, X2, X3) |
| U21(active(X1), X2, X3) | → | U21(X1, X2, X3) | | U21(X1, active(X2), X3) | → | U21(X1, X2, X3) |
| U21(X1, X2, active(X3)) | → | U21(X1, X2, X3) | | U22(mark(X1), X2, X3) | → | U22(X1, X2, X3) |
| U22(X1, mark(X2), X3) | → | U22(X1, X2, X3) | | U22(X1, X2, mark(X3)) | → | U22(X1, X2, X3) |
| U22(active(X1), X2, X3) | → | U22(X1, X2, X3) | | U22(X1, active(X2), X3) | → | U22(X1, X2, X3) |
| U22(X1, X2, active(X3)) | → | U22(X1, X2, X3) | | x(mark(X1), X2) | → | x(X1, X2) |
| x(X1, mark(X2)) | → | x(X1, X2) | | x(active(X1), X2) | → | x(X1, X2) |
| x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, tt, active, U11, mark, U12, U21, U22, x
Strategy
The following SCCs where found
| x#(mark(X1), X2) → x#(X1, X2) | x#(X1, active(X2)) → x#(X1, X2) |
| x#(X1, mark(X2)) → x#(X1, X2) | x#(active(X1), X2) → x#(X1, X2) |
| s#(mark(X)) → s#(X) | s#(active(X)) → s#(X) |
| U22#(X1, active(X2), X3) → U22#(X1, X2, X3) | U22#(active(X1), X2, X3) → U22#(X1, X2, X3) |
| U22#(mark(X1), X2, X3) → U22#(X1, X2, X3) | U22#(X1, X2, active(X3)) → U22#(X1, X2, X3) |
| U22#(X1, X2, mark(X3)) → U22#(X1, X2, X3) | U22#(X1, mark(X2), X3) → U22#(X1, X2, X3) |
| active#(U11(tt, M, N)) → mark#(U12(tt, M, N)) | active#(U22(tt, M, N)) → mark#(plus(x(N, M), N)) |
| active#(U21(tt, M, N)) → mark#(U22(tt, M, N)) | mark#(tt) → active#(tt) |
| active#(plus(N, s(M))) → mark#(U11(tt, M, N)) | active#(U12(tt, M, N)) → mark#(s(plus(N, M))) |
| mark#(U11(X1, X2, X3)) → active#(U11(mark(X1), X2, X3)) | active#(x(N, s(M))) → mark#(U21(tt, M, N)) |
| mark#(plus(X1, X2)) → mark#(X2) | mark#(U12(X1, X2, X3)) → active#(U12(mark(X1), X2, X3)) |
| mark#(s(X)) → mark#(X) | mark#(U12(X1, X2, X3)) → mark#(X1) |
| mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2))) | mark#(0) → active#(0) |
| mark#(s(X)) → active#(s(mark(X))) | mark#(x(X1, X2)) → active#(x(mark(X1), mark(X2))) |
| active#(plus(N, 0)) → mark#(N) | mark#(U22(X1, X2, X3)) → mark#(X1) |
| mark#(U22(X1, X2, X3)) → active#(U22(mark(X1), X2, X3)) | active#(x(N, 0)) → mark#(0) |
| mark#(plus(X1, X2)) → mark#(X1) | mark#(x(X1, X2)) → mark#(X2) |
| mark#(U21(X1, X2, X3)) → mark#(X1) | mark#(U21(X1, X2, X3)) → active#(U21(mark(X1), X2, X3)) |
| mark#(U11(X1, X2, X3)) → mark#(X1) | mark#(x(X1, X2)) → mark#(X1) |
| U12#(X1, active(X2), X3) → U12#(X1, X2, X3) | U12#(X1, X2, active(X3)) → U12#(X1, X2, X3) |
| U12#(mark(X1), X2, X3) → U12#(X1, X2, X3) | U12#(X1, X2, mark(X3)) → U12#(X1, X2, X3) |
| U12#(active(X1), X2, X3) → U12#(X1, X2, X3) | U12#(X1, mark(X2), X3) → U12#(X1, X2, X3) |
| plus#(X1, mark(X2)) → plus#(X1, X2) | plus#(X1, active(X2)) → plus#(X1, X2) |
| plus#(mark(X1), X2) → plus#(X1, X2) | plus#(active(X1), X2) → plus#(X1, X2) |
| U11#(X1, X2, mark(X3)) → U11#(X1, X2, X3) | U11#(X1, active(X2), X3) → U11#(X1, X2, X3) |
| U11#(active(X1), X2, X3) → U11#(X1, X2, X3) | U11#(X1, mark(X2), X3) → U11#(X1, X2, X3) |
| U11#(mark(X1), X2, X3) → U11#(X1, X2, X3) | U11#(X1, X2, active(X3)) → U11#(X1, X2, X3) |
| U21#(active(X1), X2, X3) → U21#(X1, X2, X3) | U21#(X1, active(X2), X3) → U21#(X1, X2, X3) |
| U21#(X1, mark(X2), X3) → U21#(X1, X2, X3) | U21#(X1, X2, mark(X3)) → U21#(X1, X2, X3) |
| U21#(X1, X2, active(X3)) → U21#(X1, X2, X3) | U21#(mark(X1), X2, X3) → U21#(X1, X2, X3) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U21#(active(X1), X2, X3) | → | U21#(X1, X2, X3) | | U21#(X1, active(X2), X3) | → | U21#(X1, X2, X3) |
| U21#(X1, mark(X2), X3) | → | U21#(X1, X2, X3) | | U21#(X1, X2, mark(X3)) | → | U21#(X1, X2, X3) |
| U21#(X1, X2, active(X3)) | → | U21#(X1, X2, X3) | | U21#(mark(X1), X2, X3) | → | U21#(X1, X2, X3) |
Rewrite Rules
| active(U11(tt, M, N)) | → | mark(U12(tt, M, N)) | | active(U12(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U21(tt, M, N)) | → | mark(U22(tt, M, N)) | | active(U22(tt, M, N)) | → | mark(plus(x(N, M), N)) |
| active(plus(N, 0)) | → | mark(N) | | active(plus(N, s(M))) | → | mark(U11(tt, M, N)) |
| active(x(N, 0)) | → | mark(0) | | active(x(N, s(M))) | → | mark(U21(tt, M, N)) |
| mark(U11(X1, X2, X3)) | → | active(U11(mark(X1), X2, X3)) | | mark(tt) | → | active(tt) |
| mark(U12(X1, X2, X3)) | → | active(U12(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U21(X1, X2, X3)) | → | active(U21(mark(X1), X2, X3)) |
| mark(U22(X1, X2, X3)) | → | active(U22(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| mark(0) | → | active(0) | | U11(mark(X1), X2, X3) | → | U11(X1, X2, X3) |
| U11(X1, mark(X2), X3) | → | U11(X1, X2, X3) | | U11(X1, X2, mark(X3)) | → | U11(X1, X2, X3) |
| U11(active(X1), X2, X3) | → | U11(X1, X2, X3) | | U11(X1, active(X2), X3) | → | U11(X1, X2, X3) |
| U11(X1, X2, active(X3)) | → | U11(X1, X2, X3) | | U12(mark(X1), X2, X3) | → | U12(X1, X2, X3) |
| U12(X1, mark(X2), X3) | → | U12(X1, X2, X3) | | U12(X1, X2, mark(X3)) | → | U12(X1, X2, X3) |
| U12(active(X1), X2, X3) | → | U12(X1, X2, X3) | | U12(X1, active(X2), X3) | → | U12(X1, X2, X3) |
| U12(X1, X2, active(X3)) | → | U12(X1, X2, X3) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | plus(mark(X1), X2) | → | plus(X1, X2) |
| plus(X1, mark(X2)) | → | plus(X1, X2) | | plus(active(X1), X2) | → | plus(X1, X2) |
| plus(X1, active(X2)) | → | plus(X1, X2) | | U21(mark(X1), X2, X3) | → | U21(X1, X2, X3) |
| U21(X1, mark(X2), X3) | → | U21(X1, X2, X3) | | U21(X1, X2, mark(X3)) | → | U21(X1, X2, X3) |
| U21(active(X1), X2, X3) | → | U21(X1, X2, X3) | | U21(X1, active(X2), X3) | → | U21(X1, X2, X3) |
| U21(X1, X2, active(X3)) | → | U21(X1, X2, X3) | | U22(mark(X1), X2, X3) | → | U22(X1, X2, X3) |
| U22(X1, mark(X2), X3) | → | U22(X1, X2, X3) | | U22(X1, X2, mark(X3)) | → | U22(X1, X2, X3) |
| U22(active(X1), X2, X3) | → | U22(X1, X2, X3) | | U22(X1, active(X2), X3) | → | U22(X1, X2, X3) |
| U22(X1, X2, active(X3)) | → | U22(X1, X2, X3) | | x(mark(X1), X2) | → | x(X1, X2) |
| x(X1, mark(X2)) | → | x(X1, X2) | | x(active(X1), X2) | → | x(X1, X2) |
| x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, tt, active, U11, mark, U12, U21, U22, x
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U21#(active(X1), X2, X3) | → | U21#(X1, X2, X3) | | U21#(mark(X1), X2, X3) | → | U21#(X1, X2, X3) |
Problem 10: ReductionPairSAT
Dependency Pair Problem
Dependency Pairs
| U21#(X1, mark(X2), X3) | → | U21#(X1, X2, X3) | | U21#(X1, active(X2), X3) | → | U21#(X1, X2, X3) |
| U21#(X1, X2, mark(X3)) | → | U21#(X1, X2, X3) | | U21#(X1, X2, active(X3)) | → | U21#(X1, X2, X3) |
Rewrite Rules
| active(U11(tt, M, N)) | → | mark(U12(tt, M, N)) | | active(U12(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U21(tt, M, N)) | → | mark(U22(tt, M, N)) | | active(U22(tt, M, N)) | → | mark(plus(x(N, M), N)) |
| active(plus(N, 0)) | → | mark(N) | | active(plus(N, s(M))) | → | mark(U11(tt, M, N)) |
| active(x(N, 0)) | → | mark(0) | | active(x(N, s(M))) | → | mark(U21(tt, M, N)) |
| mark(U11(X1, X2, X3)) | → | active(U11(mark(X1), X2, X3)) | | mark(tt) | → | active(tt) |
| mark(U12(X1, X2, X3)) | → | active(U12(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U21(X1, X2, X3)) | → | active(U21(mark(X1), X2, X3)) |
| mark(U22(X1, X2, X3)) | → | active(U22(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| mark(0) | → | active(0) | | U11(mark(X1), X2, X3) | → | U11(X1, X2, X3) |
| U11(X1, mark(X2), X3) | → | U11(X1, X2, X3) | | U11(X1, X2, mark(X3)) | → | U11(X1, X2, X3) |
| U11(active(X1), X2, X3) | → | U11(X1, X2, X3) | | U11(X1, active(X2), X3) | → | U11(X1, X2, X3) |
| U11(X1, X2, active(X3)) | → | U11(X1, X2, X3) | | U12(mark(X1), X2, X3) | → | U12(X1, X2, X3) |
| U12(X1, mark(X2), X3) | → | U12(X1, X2, X3) | | U12(X1, X2, mark(X3)) | → | U12(X1, X2, X3) |
| U12(active(X1), X2, X3) | → | U12(X1, X2, X3) | | U12(X1, active(X2), X3) | → | U12(X1, X2, X3) |
| U12(X1, X2, active(X3)) | → | U12(X1, X2, X3) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | plus(mark(X1), X2) | → | plus(X1, X2) |
| plus(X1, mark(X2)) | → | plus(X1, X2) | | plus(active(X1), X2) | → | plus(X1, X2) |
| plus(X1, active(X2)) | → | plus(X1, X2) | | U21(mark(X1), X2, X3) | → | U21(X1, X2, X3) |
| U21(X1, mark(X2), X3) | → | U21(X1, X2, X3) | | U21(X1, X2, mark(X3)) | → | U21(X1, X2, X3) |
| U21(active(X1), X2, X3) | → | U21(X1, X2, X3) | | U21(X1, active(X2), X3) | → | U21(X1, X2, X3) |
| U21(X1, X2, active(X3)) | → | U21(X1, X2, X3) | | U22(mark(X1), X2, X3) | → | U22(X1, X2, X3) |
| U22(X1, mark(X2), X3) | → | U22(X1, X2, X3) | | U22(X1, X2, mark(X3)) | → | U22(X1, X2, X3) |
| U22(active(X1), X2, X3) | → | U22(X1, X2, X3) | | U22(X1, active(X2), X3) | → | U22(X1, X2, X3) |
| U22(X1, X2, active(X3)) | → | U22(X1, X2, X3) | | x(mark(X1), X2) | → | x(X1, X2) |
| x(X1, mark(X2)) | → | x(X1, X2) | | x(active(X1), X2) | → | x(X1, X2) |
| x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, tt, U11, active, U12, mark, U21, x, U22
Strategy
Function Precedence
active = mark < plus = 0 = s = tt = U21# = U11 = U12 = U21 = x = U22
Argument Filtering
plus: all arguments are removed from plus
0: all arguments are removed from 0
s: all arguments are removed from s
tt: all arguments are removed from tt
U21#: collapses to 2
U11: 1 2 3
active: collapses to 1
mark: 1
U12: collapses to 2
U21: all arguments are removed from U21
x: all arguments are removed from x
U22: 2
Status
plus: multiset
0: multiset
s: multiset
tt: multiset
U11: lexicographic with permutation 1 → 2 2 → 1 3 → 3
mark: multiset
U21: multiset
x: multiset
U22: lexicographic with permutation 2 → 1
Usable Rules
There are no usable rules.
The dependency pairs and usable rules are stronlgy conservative!
Eliminated dependency pairs
The following dependency pairs (at least) can be eliminated according to the given precedence.
| U21#(X1, mark(X2), X3) → U21#(X1, X2, X3) |
Problem 17: ReductionPairSAT
Dependency Pair Problem
Dependency Pairs
| U21#(X1, active(X2), X3) | → | U21#(X1, X2, X3) | | U21#(X1, X2, mark(X3)) | → | U21#(X1, X2, X3) |
| U21#(X1, X2, active(X3)) | → | U21#(X1, X2, X3) |
Rewrite Rules
| active(U11(tt, M, N)) | → | mark(U12(tt, M, N)) | | active(U12(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U21(tt, M, N)) | → | mark(U22(tt, M, N)) | | active(U22(tt, M, N)) | → | mark(plus(x(N, M), N)) |
| active(plus(N, 0)) | → | mark(N) | | active(plus(N, s(M))) | → | mark(U11(tt, M, N)) |
| active(x(N, 0)) | → | mark(0) | | active(x(N, s(M))) | → | mark(U21(tt, M, N)) |
| mark(U11(X1, X2, X3)) | → | active(U11(mark(X1), X2, X3)) | | mark(tt) | → | active(tt) |
| mark(U12(X1, X2, X3)) | → | active(U12(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U21(X1, X2, X3)) | → | active(U21(mark(X1), X2, X3)) |
| mark(U22(X1, X2, X3)) | → | active(U22(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| mark(0) | → | active(0) | | U11(mark(X1), X2, X3) | → | U11(X1, X2, X3) |
| U11(X1, mark(X2), X3) | → | U11(X1, X2, X3) | | U11(X1, X2, mark(X3)) | → | U11(X1, X2, X3) |
| U11(active(X1), X2, X3) | → | U11(X1, X2, X3) | | U11(X1, active(X2), X3) | → | U11(X1, X2, X3) |
| U11(X1, X2, active(X3)) | → | U11(X1, X2, X3) | | U12(mark(X1), X2, X3) | → | U12(X1, X2, X3) |
| U12(X1, mark(X2), X3) | → | U12(X1, X2, X3) | | U12(X1, X2, mark(X3)) | → | U12(X1, X2, X3) |
| U12(active(X1), X2, X3) | → | U12(X1, X2, X3) | | U12(X1, active(X2), X3) | → | U12(X1, X2, X3) |
| U12(X1, X2, active(X3)) | → | U12(X1, X2, X3) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | plus(mark(X1), X2) | → | plus(X1, X2) |
| plus(X1, mark(X2)) | → | plus(X1, X2) | | plus(active(X1), X2) | → | plus(X1, X2) |
| plus(X1, active(X2)) | → | plus(X1, X2) | | U21(mark(X1), X2, X3) | → | U21(X1, X2, X3) |
| U21(X1, mark(X2), X3) | → | U21(X1, X2, X3) | | U21(X1, X2, mark(X3)) | → | U21(X1, X2, X3) |
| U21(active(X1), X2, X3) | → | U21(X1, X2, X3) | | U21(X1, active(X2), X3) | → | U21(X1, X2, X3) |
| U21(X1, X2, active(X3)) | → | U21(X1, X2, X3) | | U22(mark(X1), X2, X3) | → | U22(X1, X2, X3) |
| U22(X1, mark(X2), X3) | → | U22(X1, X2, X3) | | U22(X1, X2, mark(X3)) | → | U22(X1, X2, X3) |
| U22(active(X1), X2, X3) | → | U22(X1, X2, X3) | | U22(X1, active(X2), X3) | → | U22(X1, X2, X3) |
| U22(X1, X2, active(X3)) | → | U22(X1, X2, X3) | | x(mark(X1), X2) | → | x(X1, X2) |
| x(X1, mark(X2)) | → | x(X1, X2) | | x(active(X1), X2) | → | x(X1, X2) |
| x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, tt, active, U11, mark, U12, U21, U22, x
Strategy
Function Precedence
active < plus = 0 = s = tt = U21# = U11 = mark = U12 = U21 = x = U22
Argument Filtering
plus: all arguments are removed from plus
0: all arguments are removed from 0
s: all arguments are removed from s
tt: all arguments are removed from tt
U21#: collapses to 3
U11: 3
active: 1
mark: collapses to 1
U12: 1
U21: all arguments are removed from U21
x: collapses to 2
U22: 3
Status
plus: multiset
0: multiset
s: multiset
tt: multiset
U11: lexicographic with permutation 3 → 1
active: multiset
U12: lexicographic with permutation 1 → 1
U21: multiset
U22: lexicographic with permutation 3 → 1
Usable Rules
There are no usable rules.
The dependency pairs and usable rules are stronlgy conservative!
Eliminated dependency pairs
The following dependency pairs (at least) can be eliminated according to the given precedence.
| U21#(X1, X2, active(X3)) → U21#(X1, X2, X3) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U22#(X1, active(X2), X3) | → | U22#(X1, X2, X3) | | U22#(active(X1), X2, X3) | → | U22#(X1, X2, X3) |
| U22#(mark(X1), X2, X3) | → | U22#(X1, X2, X3) | | U22#(X1, X2, active(X3)) | → | U22#(X1, X2, X3) |
| U22#(X1, X2, mark(X3)) | → | U22#(X1, X2, X3) | | U22#(X1, mark(X2), X3) | → | U22#(X1, X2, X3) |
Rewrite Rules
| active(U11(tt, M, N)) | → | mark(U12(tt, M, N)) | | active(U12(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U21(tt, M, N)) | → | mark(U22(tt, M, N)) | | active(U22(tt, M, N)) | → | mark(plus(x(N, M), N)) |
| active(plus(N, 0)) | → | mark(N) | | active(plus(N, s(M))) | → | mark(U11(tt, M, N)) |
| active(x(N, 0)) | → | mark(0) | | active(x(N, s(M))) | → | mark(U21(tt, M, N)) |
| mark(U11(X1, X2, X3)) | → | active(U11(mark(X1), X2, X3)) | | mark(tt) | → | active(tt) |
| mark(U12(X1, X2, X3)) | → | active(U12(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U21(X1, X2, X3)) | → | active(U21(mark(X1), X2, X3)) |
| mark(U22(X1, X2, X3)) | → | active(U22(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| mark(0) | → | active(0) | | U11(mark(X1), X2, X3) | → | U11(X1, X2, X3) |
| U11(X1, mark(X2), X3) | → | U11(X1, X2, X3) | | U11(X1, X2, mark(X3)) | → | U11(X1, X2, X3) |
| U11(active(X1), X2, X3) | → | U11(X1, X2, X3) | | U11(X1, active(X2), X3) | → | U11(X1, X2, X3) |
| U11(X1, X2, active(X3)) | → | U11(X1, X2, X3) | | U12(mark(X1), X2, X3) | → | U12(X1, X2, X3) |
| U12(X1, mark(X2), X3) | → | U12(X1, X2, X3) | | U12(X1, X2, mark(X3)) | → | U12(X1, X2, X3) |
| U12(active(X1), X2, X3) | → | U12(X1, X2, X3) | | U12(X1, active(X2), X3) | → | U12(X1, X2, X3) |
| U12(X1, X2, active(X3)) | → | U12(X1, X2, X3) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | plus(mark(X1), X2) | → | plus(X1, X2) |
| plus(X1, mark(X2)) | → | plus(X1, X2) | | plus(active(X1), X2) | → | plus(X1, X2) |
| plus(X1, active(X2)) | → | plus(X1, X2) | | U21(mark(X1), X2, X3) | → | U21(X1, X2, X3) |
| U21(X1, mark(X2), X3) | → | U21(X1, X2, X3) | | U21(X1, X2, mark(X3)) | → | U21(X1, X2, X3) |
| U21(active(X1), X2, X3) | → | U21(X1, X2, X3) | | U21(X1, active(X2), X3) | → | U21(X1, X2, X3) |
| U21(X1, X2, active(X3)) | → | U21(X1, X2, X3) | | U22(mark(X1), X2, X3) | → | U22(X1, X2, X3) |
| U22(X1, mark(X2), X3) | → | U22(X1, X2, X3) | | U22(X1, X2, mark(X3)) | → | U22(X1, X2, X3) |
| U22(active(X1), X2, X3) | → | U22(X1, X2, X3) | | U22(X1, active(X2), X3) | → | U22(X1, X2, X3) |
| U22(X1, X2, active(X3)) | → | U22(X1, X2, X3) | | x(mark(X1), X2) | → | x(X1, X2) |
| x(X1, mark(X2)) | → | x(X1, X2) | | x(active(X1), X2) | → | x(X1, X2) |
| x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, tt, active, U11, mark, U12, U21, U22, x
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U22#(active(X1), X2, X3) | → | U22#(X1, X2, X3) | | U22#(mark(X1), X2, X3) | → | U22#(X1, X2, X3) |
Problem 11: ReductionPairSAT
Dependency Pair Problem
Dependency Pairs
| U22#(X1, active(X2), X3) | → | U22#(X1, X2, X3) | | U22#(X1, X2, active(X3)) | → | U22#(X1, X2, X3) |
| U22#(X1, X2, mark(X3)) | → | U22#(X1, X2, X3) | | U22#(X1, mark(X2), X3) | → | U22#(X1, X2, X3) |
Rewrite Rules
| active(U11(tt, M, N)) | → | mark(U12(tt, M, N)) | | active(U12(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U21(tt, M, N)) | → | mark(U22(tt, M, N)) | | active(U22(tt, M, N)) | → | mark(plus(x(N, M), N)) |
| active(plus(N, 0)) | → | mark(N) | | active(plus(N, s(M))) | → | mark(U11(tt, M, N)) |
| active(x(N, 0)) | → | mark(0) | | active(x(N, s(M))) | → | mark(U21(tt, M, N)) |
| mark(U11(X1, X2, X3)) | → | active(U11(mark(X1), X2, X3)) | | mark(tt) | → | active(tt) |
| mark(U12(X1, X2, X3)) | → | active(U12(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U21(X1, X2, X3)) | → | active(U21(mark(X1), X2, X3)) |
| mark(U22(X1, X2, X3)) | → | active(U22(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| mark(0) | → | active(0) | | U11(mark(X1), X2, X3) | → | U11(X1, X2, X3) |
| U11(X1, mark(X2), X3) | → | U11(X1, X2, X3) | | U11(X1, X2, mark(X3)) | → | U11(X1, X2, X3) |
| U11(active(X1), X2, X3) | → | U11(X1, X2, X3) | | U11(X1, active(X2), X3) | → | U11(X1, X2, X3) |
| U11(X1, X2, active(X3)) | → | U11(X1, X2, X3) | | U12(mark(X1), X2, X3) | → | U12(X1, X2, X3) |
| U12(X1, mark(X2), X3) | → | U12(X1, X2, X3) | | U12(X1, X2, mark(X3)) | → | U12(X1, X2, X3) |
| U12(active(X1), X2, X3) | → | U12(X1, X2, X3) | | U12(X1, active(X2), X3) | → | U12(X1, X2, X3) |
| U12(X1, X2, active(X3)) | → | U12(X1, X2, X3) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | plus(mark(X1), X2) | → | plus(X1, X2) |
| plus(X1, mark(X2)) | → | plus(X1, X2) | | plus(active(X1), X2) | → | plus(X1, X2) |
| plus(X1, active(X2)) | → | plus(X1, X2) | | U21(mark(X1), X2, X3) | → | U21(X1, X2, X3) |
| U21(X1, mark(X2), X3) | → | U21(X1, X2, X3) | | U21(X1, X2, mark(X3)) | → | U21(X1, X2, X3) |
| U21(active(X1), X2, X3) | → | U21(X1, X2, X3) | | U21(X1, active(X2), X3) | → | U21(X1, X2, X3) |
| U21(X1, X2, active(X3)) | → | U21(X1, X2, X3) | | U22(mark(X1), X2, X3) | → | U22(X1, X2, X3) |
| U22(X1, mark(X2), X3) | → | U22(X1, X2, X3) | | U22(X1, X2, mark(X3)) | → | U22(X1, X2, X3) |
| U22(active(X1), X2, X3) | → | U22(X1, X2, X3) | | U22(X1, active(X2), X3) | → | U22(X1, X2, X3) |
| U22(X1, X2, active(X3)) | → | U22(X1, X2, X3) | | x(mark(X1), X2) | → | x(X1, X2) |
| x(X1, mark(X2)) | → | x(X1, X2) | | x(active(X1), X2) | → | x(X1, X2) |
| x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, tt, U11, active, U12, mark, U21, x, U22
Strategy
Function Precedence
mark < plus = 0 = s = tt = U22# = U11 = active = U12 = U21 = x = U22
Argument Filtering
plus: all arguments are removed from plus
0: all arguments are removed from 0
s: 1
tt: all arguments are removed from tt
U22#: collapses to 3
U11: 1 2
active: 1
mark: collapses to 1
U12: 2
U21: 1 2 3
x: 2
U22: 2 3
Status
plus: multiset
0: multiset
s: lexicographic with permutation 1 → 1
tt: multiset
U11: lexicographic with permutation 1 → 2 2 → 1
active: multiset
U12: lexicographic with permutation 2 → 1
U21: lexicographic with permutation 1 → 2 2 → 1 3 → 3
x: lexicographic with permutation 2 → 1
U22: lexicographic with permutation 2 → 1 3 → 2
Usable Rules
There are no usable rules.
The dependency pairs and usable rules are stronlgy conservative!
Eliminated dependency pairs
The following dependency pairs (at least) can be eliminated according to the given precedence.
| U22#(X1, X2, active(X3)) → U22#(X1, X2, X3) |
Problem 4: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| x#(mark(X1), X2) | → | x#(X1, X2) | | x#(X1, active(X2)) | → | x#(X1, X2) |
| x#(X1, mark(X2)) | → | x#(X1, X2) | | x#(active(X1), X2) | → | x#(X1, X2) |
Rewrite Rules
| active(U11(tt, M, N)) | → | mark(U12(tt, M, N)) | | active(U12(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U21(tt, M, N)) | → | mark(U22(tt, M, N)) | | active(U22(tt, M, N)) | → | mark(plus(x(N, M), N)) |
| active(plus(N, 0)) | → | mark(N) | | active(plus(N, s(M))) | → | mark(U11(tt, M, N)) |
| active(x(N, 0)) | → | mark(0) | | active(x(N, s(M))) | → | mark(U21(tt, M, N)) |
| mark(U11(X1, X2, X3)) | → | active(U11(mark(X1), X2, X3)) | | mark(tt) | → | active(tt) |
| mark(U12(X1, X2, X3)) | → | active(U12(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U21(X1, X2, X3)) | → | active(U21(mark(X1), X2, X3)) |
| mark(U22(X1, X2, X3)) | → | active(U22(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| mark(0) | → | active(0) | | U11(mark(X1), X2, X3) | → | U11(X1, X2, X3) |
| U11(X1, mark(X2), X3) | → | U11(X1, X2, X3) | | U11(X1, X2, mark(X3)) | → | U11(X1, X2, X3) |
| U11(active(X1), X2, X3) | → | U11(X1, X2, X3) | | U11(X1, active(X2), X3) | → | U11(X1, X2, X3) |
| U11(X1, X2, active(X3)) | → | U11(X1, X2, X3) | | U12(mark(X1), X2, X3) | → | U12(X1, X2, X3) |
| U12(X1, mark(X2), X3) | → | U12(X1, X2, X3) | | U12(X1, X2, mark(X3)) | → | U12(X1, X2, X3) |
| U12(active(X1), X2, X3) | → | U12(X1, X2, X3) | | U12(X1, active(X2), X3) | → | U12(X1, X2, X3) |
| U12(X1, X2, active(X3)) | → | U12(X1, X2, X3) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | plus(mark(X1), X2) | → | plus(X1, X2) |
| plus(X1, mark(X2)) | → | plus(X1, X2) | | plus(active(X1), X2) | → | plus(X1, X2) |
| plus(X1, active(X2)) | → | plus(X1, X2) | | U21(mark(X1), X2, X3) | → | U21(X1, X2, X3) |
| U21(X1, mark(X2), X3) | → | U21(X1, X2, X3) | | U21(X1, X2, mark(X3)) | → | U21(X1, X2, X3) |
| U21(active(X1), X2, X3) | → | U21(X1, X2, X3) | | U21(X1, active(X2), X3) | → | U21(X1, X2, X3) |
| U21(X1, X2, active(X3)) | → | U21(X1, X2, X3) | | U22(mark(X1), X2, X3) | → | U22(X1, X2, X3) |
| U22(X1, mark(X2), X3) | → | U22(X1, X2, X3) | | U22(X1, X2, mark(X3)) | → | U22(X1, X2, X3) |
| U22(active(X1), X2, X3) | → | U22(X1, X2, X3) | | U22(X1, active(X2), X3) | → | U22(X1, X2, X3) |
| U22(X1, X2, active(X3)) | → | U22(X1, X2, X3) | | x(mark(X1), X2) | → | x(X1, X2) |
| x(X1, mark(X2)) | → | x(X1, X2) | | x(active(X1), X2) | → | x(X1, X2) |
| x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, tt, active, U11, mark, U12, U21, U22, x
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| x#(mark(X1), X2) | → | x#(X1, X2) | | x#(active(X1), X2) | → | x#(X1, X2) |
Problem 12: ReductionPairSAT
Dependency Pair Problem
Dependency Pairs
| x#(X1, active(X2)) | → | x#(X1, X2) | | x#(X1, mark(X2)) | → | x#(X1, X2) |
Rewrite Rules
| active(U11(tt, M, N)) | → | mark(U12(tt, M, N)) | | active(U12(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U21(tt, M, N)) | → | mark(U22(tt, M, N)) | | active(U22(tt, M, N)) | → | mark(plus(x(N, M), N)) |
| active(plus(N, 0)) | → | mark(N) | | active(plus(N, s(M))) | → | mark(U11(tt, M, N)) |
| active(x(N, 0)) | → | mark(0) | | active(x(N, s(M))) | → | mark(U21(tt, M, N)) |
| mark(U11(X1, X2, X3)) | → | active(U11(mark(X1), X2, X3)) | | mark(tt) | → | active(tt) |
| mark(U12(X1, X2, X3)) | → | active(U12(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U21(X1, X2, X3)) | → | active(U21(mark(X1), X2, X3)) |
| mark(U22(X1, X2, X3)) | → | active(U22(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| mark(0) | → | active(0) | | U11(mark(X1), X2, X3) | → | U11(X1, X2, X3) |
| U11(X1, mark(X2), X3) | → | U11(X1, X2, X3) | | U11(X1, X2, mark(X3)) | → | U11(X1, X2, X3) |
| U11(active(X1), X2, X3) | → | U11(X1, X2, X3) | | U11(X1, active(X2), X3) | → | U11(X1, X2, X3) |
| U11(X1, X2, active(X3)) | → | U11(X1, X2, X3) | | U12(mark(X1), X2, X3) | → | U12(X1, X2, X3) |
| U12(X1, mark(X2), X3) | → | U12(X1, X2, X3) | | U12(X1, X2, mark(X3)) | → | U12(X1, X2, X3) |
| U12(active(X1), X2, X3) | → | U12(X1, X2, X3) | | U12(X1, active(X2), X3) | → | U12(X1, X2, X3) |
| U12(X1, X2, active(X3)) | → | U12(X1, X2, X3) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | plus(mark(X1), X2) | → | plus(X1, X2) |
| plus(X1, mark(X2)) | → | plus(X1, X2) | | plus(active(X1), X2) | → | plus(X1, X2) |
| plus(X1, active(X2)) | → | plus(X1, X2) | | U21(mark(X1), X2, X3) | → | U21(X1, X2, X3) |
| U21(X1, mark(X2), X3) | → | U21(X1, X2, X3) | | U21(X1, X2, mark(X3)) | → | U21(X1, X2, X3) |
| U21(active(X1), X2, X3) | → | U21(X1, X2, X3) | | U21(X1, active(X2), X3) | → | U21(X1, X2, X3) |
| U21(X1, X2, active(X3)) | → | U21(X1, X2, X3) | | U22(mark(X1), X2, X3) | → | U22(X1, X2, X3) |
| U22(X1, mark(X2), X3) | → | U22(X1, X2, X3) | | U22(X1, X2, mark(X3)) | → | U22(X1, X2, X3) |
| U22(active(X1), X2, X3) | → | U22(X1, X2, X3) | | U22(X1, active(X2), X3) | → | U22(X1, X2, X3) |
| U22(X1, X2, active(X3)) | → | U22(X1, X2, X3) | | x(mark(X1), X2) | → | x(X1, X2) |
| x(X1, mark(X2)) | → | x(X1, X2) | | x(active(X1), X2) | → | x(X1, X2) |
| x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, tt, U11, active, U12, mark, U21, x, U22
Strategy
Function Precedence
x# < active = mark < plus = 0 = s = tt = U11 = U12 = U21 = x = U22
Argument Filtering
plus: all arguments are removed from plus
0: all arguments are removed from 0
s: all arguments are removed from s
tt: all arguments are removed from tt
U11: all arguments are removed from U11
active: 1
mark: collapses to 1
U12: all arguments are removed from U12
x#: collapses to 2
U21: all arguments are removed from U21
x: 1 2
U22: all arguments are removed from U22
Status
plus: multiset
0: multiset
s: multiset
tt: multiset
U11: multiset
active: multiset
U12: multiset
U21: multiset
x: lexicographic with permutation 1 → 1 2 → 2
U22: multiset
Usable Rules
There are no usable rules.
The dependency pairs and usable rules are stronlgy conservative!
Eliminated dependency pairs
The following dependency pairs (at least) can be eliminated according to the given precedence.
| x#(X1, active(X2)) → x#(X1, X2) |
Problem 5: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U12#(X1, active(X2), X3) | → | U12#(X1, X2, X3) | | U12#(X1, X2, active(X3)) | → | U12#(X1, X2, X3) |
| U12#(mark(X1), X2, X3) | → | U12#(X1, X2, X3) | | U12#(X1, X2, mark(X3)) | → | U12#(X1, X2, X3) |
| U12#(active(X1), X2, X3) | → | U12#(X1, X2, X3) | | U12#(X1, mark(X2), X3) | → | U12#(X1, X2, X3) |
Rewrite Rules
| active(U11(tt, M, N)) | → | mark(U12(tt, M, N)) | | active(U12(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U21(tt, M, N)) | → | mark(U22(tt, M, N)) | | active(U22(tt, M, N)) | → | mark(plus(x(N, M), N)) |
| active(plus(N, 0)) | → | mark(N) | | active(plus(N, s(M))) | → | mark(U11(tt, M, N)) |
| active(x(N, 0)) | → | mark(0) | | active(x(N, s(M))) | → | mark(U21(tt, M, N)) |
| mark(U11(X1, X2, X3)) | → | active(U11(mark(X1), X2, X3)) | | mark(tt) | → | active(tt) |
| mark(U12(X1, X2, X3)) | → | active(U12(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U21(X1, X2, X3)) | → | active(U21(mark(X1), X2, X3)) |
| mark(U22(X1, X2, X3)) | → | active(U22(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| mark(0) | → | active(0) | | U11(mark(X1), X2, X3) | → | U11(X1, X2, X3) |
| U11(X1, mark(X2), X3) | → | U11(X1, X2, X3) | | U11(X1, X2, mark(X3)) | → | U11(X1, X2, X3) |
| U11(active(X1), X2, X3) | → | U11(X1, X2, X3) | | U11(X1, active(X2), X3) | → | U11(X1, X2, X3) |
| U11(X1, X2, active(X3)) | → | U11(X1, X2, X3) | | U12(mark(X1), X2, X3) | → | U12(X1, X2, X3) |
| U12(X1, mark(X2), X3) | → | U12(X1, X2, X3) | | U12(X1, X2, mark(X3)) | → | U12(X1, X2, X3) |
| U12(active(X1), X2, X3) | → | U12(X1, X2, X3) | | U12(X1, active(X2), X3) | → | U12(X1, X2, X3) |
| U12(X1, X2, active(X3)) | → | U12(X1, X2, X3) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | plus(mark(X1), X2) | → | plus(X1, X2) |
| plus(X1, mark(X2)) | → | plus(X1, X2) | | plus(active(X1), X2) | → | plus(X1, X2) |
| plus(X1, active(X2)) | → | plus(X1, X2) | | U21(mark(X1), X2, X3) | → | U21(X1, X2, X3) |
| U21(X1, mark(X2), X3) | → | U21(X1, X2, X3) | | U21(X1, X2, mark(X3)) | → | U21(X1, X2, X3) |
| U21(active(X1), X2, X3) | → | U21(X1, X2, X3) | | U21(X1, active(X2), X3) | → | U21(X1, X2, X3) |
| U21(X1, X2, active(X3)) | → | U21(X1, X2, X3) | | U22(mark(X1), X2, X3) | → | U22(X1, X2, X3) |
| U22(X1, mark(X2), X3) | → | U22(X1, X2, X3) | | U22(X1, X2, mark(X3)) | → | U22(X1, X2, X3) |
| U22(active(X1), X2, X3) | → | U22(X1, X2, X3) | | U22(X1, active(X2), X3) | → | U22(X1, X2, X3) |
| U22(X1, X2, active(X3)) | → | U22(X1, X2, X3) | | x(mark(X1), X2) | → | x(X1, X2) |
| x(X1, mark(X2)) | → | x(X1, X2) | | x(active(X1), X2) | → | x(X1, X2) |
| x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, tt, active, U11, mark, U12, U21, U22, x
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U12#(active(X1), X2, X3) | → | U12#(X1, X2, X3) | | U12#(mark(X1), X2, X3) | → | U12#(X1, X2, X3) |
Problem 13: ReductionPairSAT
Dependency Pair Problem
Dependency Pairs
| U12#(X1, active(X2), X3) | → | U12#(X1, X2, X3) | | U12#(X1, X2, active(X3)) | → | U12#(X1, X2, X3) |
| U12#(X1, X2, mark(X3)) | → | U12#(X1, X2, X3) | | U12#(X1, mark(X2), X3) | → | U12#(X1, X2, X3) |
Rewrite Rules
| active(U11(tt, M, N)) | → | mark(U12(tt, M, N)) | | active(U12(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U21(tt, M, N)) | → | mark(U22(tt, M, N)) | | active(U22(tt, M, N)) | → | mark(plus(x(N, M), N)) |
| active(plus(N, 0)) | → | mark(N) | | active(plus(N, s(M))) | → | mark(U11(tt, M, N)) |
| active(x(N, 0)) | → | mark(0) | | active(x(N, s(M))) | → | mark(U21(tt, M, N)) |
| mark(U11(X1, X2, X3)) | → | active(U11(mark(X1), X2, X3)) | | mark(tt) | → | active(tt) |
| mark(U12(X1, X2, X3)) | → | active(U12(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U21(X1, X2, X3)) | → | active(U21(mark(X1), X2, X3)) |
| mark(U22(X1, X2, X3)) | → | active(U22(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| mark(0) | → | active(0) | | U11(mark(X1), X2, X3) | → | U11(X1, X2, X3) |
| U11(X1, mark(X2), X3) | → | U11(X1, X2, X3) | | U11(X1, X2, mark(X3)) | → | U11(X1, X2, X3) |
| U11(active(X1), X2, X3) | → | U11(X1, X2, X3) | | U11(X1, active(X2), X3) | → | U11(X1, X2, X3) |
| U11(X1, X2, active(X3)) | → | U11(X1, X2, X3) | | U12(mark(X1), X2, X3) | → | U12(X1, X2, X3) |
| U12(X1, mark(X2), X3) | → | U12(X1, X2, X3) | | U12(X1, X2, mark(X3)) | → | U12(X1, X2, X3) |
| U12(active(X1), X2, X3) | → | U12(X1, X2, X3) | | U12(X1, active(X2), X3) | → | U12(X1, X2, X3) |
| U12(X1, X2, active(X3)) | → | U12(X1, X2, X3) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | plus(mark(X1), X2) | → | plus(X1, X2) |
| plus(X1, mark(X2)) | → | plus(X1, X2) | | plus(active(X1), X2) | → | plus(X1, X2) |
| plus(X1, active(X2)) | → | plus(X1, X2) | | U21(mark(X1), X2, X3) | → | U21(X1, X2, X3) |
| U21(X1, mark(X2), X3) | → | U21(X1, X2, X3) | | U21(X1, X2, mark(X3)) | → | U21(X1, X2, X3) |
| U21(active(X1), X2, X3) | → | U21(X1, X2, X3) | | U21(X1, active(X2), X3) | → | U21(X1, X2, X3) |
| U21(X1, X2, active(X3)) | → | U21(X1, X2, X3) | | U22(mark(X1), X2, X3) | → | U22(X1, X2, X3) |
| U22(X1, mark(X2), X3) | → | U22(X1, X2, X3) | | U22(X1, X2, mark(X3)) | → | U22(X1, X2, X3) |
| U22(active(X1), X2, X3) | → | U22(X1, X2, X3) | | U22(X1, active(X2), X3) | → | U22(X1, X2, X3) |
| U22(X1, X2, active(X3)) | → | U22(X1, X2, X3) | | x(mark(X1), X2) | → | x(X1, X2) |
| x(X1, mark(X2)) | → | x(X1, X2) | | x(active(X1), X2) | → | x(X1, X2) |
| x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, tt, U11, active, U12, mark, U21, x, U22
Strategy
Function Precedence
mark < plus = U12# = 0 = s = tt = U11 = active = U12 = U21 = x = U22
Argument Filtering
plus: 1
U12#: collapses to 3
0: all arguments are removed from 0
s: collapses to 1
tt: all arguments are removed from tt
U11: all arguments are removed from U11
active: 1
mark: collapses to 1
U12: collapses to 1
U21: 1 2 3
x: 1
U22: all arguments are removed from U22
Status
plus: lexicographic with permutation 1 → 1
0: multiset
tt: multiset
U11: multiset
active: multiset
U21: lexicographic with permutation 1 → 2 2 → 3 3 → 1
x: lexicographic with permutation 1 → 1
U22: multiset
Usable Rules
There are no usable rules.
The dependency pairs and usable rules are stronlgy conservative!
Eliminated dependency pairs
The following dependency pairs (at least) can be eliminated according to the given precedence.
| U12#(X1, X2, active(X3)) → U12#(X1, X2, X3) |
Problem 6: ReductionPairSAT
Dependency Pair Problem
Dependency Pairs
| active#(U11(tt, M, N)) | → | mark#(U12(tt, M, N)) | | active#(U22(tt, M, N)) | → | mark#(plus(x(N, M), N)) |
| active#(U21(tt, M, N)) | → | mark#(U22(tt, M, N)) | | mark#(tt) | → | active#(tt) |
| active#(plus(N, s(M))) | → | mark#(U11(tt, M, N)) | | active#(U12(tt, M, N)) | → | mark#(s(plus(N, M))) |
| mark#(U11(X1, X2, X3)) | → | active#(U11(mark(X1), X2, X3)) | | active#(x(N, s(M))) | → | mark#(U21(tt, M, N)) |
| mark#(plus(X1, X2)) | → | mark#(X2) | | mark#(U12(X1, X2, X3)) | → | active#(U12(mark(X1), X2, X3)) |
| mark#(s(X)) | → | mark#(X) | | mark#(U12(X1, X2, X3)) | → | mark#(X1) |
| mark#(0) | → | active#(0) | | mark#(plus(X1, X2)) | → | active#(plus(mark(X1), mark(X2))) |
| mark#(s(X)) | → | active#(s(mark(X))) | | mark#(x(X1, X2)) | → | active#(x(mark(X1), mark(X2))) |
| active#(plus(N, 0)) | → | mark#(N) | | mark#(U22(X1, X2, X3)) | → | mark#(X1) |
| mark#(U22(X1, X2, X3)) | → | active#(U22(mark(X1), X2, X3)) | | mark#(plus(X1, X2)) | → | mark#(X1) |
| active#(x(N, 0)) | → | mark#(0) | | mark#(x(X1, X2)) | → | mark#(X2) |
| mark#(U21(X1, X2, X3)) | → | mark#(X1) | | mark#(U21(X1, X2, X3)) | → | active#(U21(mark(X1), X2, X3)) |
| mark#(U11(X1, X2, X3)) | → | mark#(X1) | | mark#(x(X1, X2)) | → | mark#(X1) |
Rewrite Rules
| active(U11(tt, M, N)) | → | mark(U12(tt, M, N)) | | active(U12(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U21(tt, M, N)) | → | mark(U22(tt, M, N)) | | active(U22(tt, M, N)) | → | mark(plus(x(N, M), N)) |
| active(plus(N, 0)) | → | mark(N) | | active(plus(N, s(M))) | → | mark(U11(tt, M, N)) |
| active(x(N, 0)) | → | mark(0) | | active(x(N, s(M))) | → | mark(U21(tt, M, N)) |
| mark(U11(X1, X2, X3)) | → | active(U11(mark(X1), X2, X3)) | | mark(tt) | → | active(tt) |
| mark(U12(X1, X2, X3)) | → | active(U12(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U21(X1, X2, X3)) | → | active(U21(mark(X1), X2, X3)) |
| mark(U22(X1, X2, X3)) | → | active(U22(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| mark(0) | → | active(0) | | U11(mark(X1), X2, X3) | → | U11(X1, X2, X3) |
| U11(X1, mark(X2), X3) | → | U11(X1, X2, X3) | | U11(X1, X2, mark(X3)) | → | U11(X1, X2, X3) |
| U11(active(X1), X2, X3) | → | U11(X1, X2, X3) | | U11(X1, active(X2), X3) | → | U11(X1, X2, X3) |
| U11(X1, X2, active(X3)) | → | U11(X1, X2, X3) | | U12(mark(X1), X2, X3) | → | U12(X1, X2, X3) |
| U12(X1, mark(X2), X3) | → | U12(X1, X2, X3) | | U12(X1, X2, mark(X3)) | → | U12(X1, X2, X3) |
| U12(active(X1), X2, X3) | → | U12(X1, X2, X3) | | U12(X1, active(X2), X3) | → | U12(X1, X2, X3) |
| U12(X1, X2, active(X3)) | → | U12(X1, X2, X3) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | plus(mark(X1), X2) | → | plus(X1, X2) |
| plus(X1, mark(X2)) | → | plus(X1, X2) | | plus(active(X1), X2) | → | plus(X1, X2) |
| plus(X1, active(X2)) | → | plus(X1, X2) | | U21(mark(X1), X2, X3) | → | U21(X1, X2, X3) |
| U21(X1, mark(X2), X3) | → | U21(X1, X2, X3) | | U21(X1, X2, mark(X3)) | → | U21(X1, X2, X3) |
| U21(active(X1), X2, X3) | → | U21(X1, X2, X3) | | U21(X1, active(X2), X3) | → | U21(X1, X2, X3) |
| U21(X1, X2, active(X3)) | → | U21(X1, X2, X3) | | U22(mark(X1), X2, X3) | → | U22(X1, X2, X3) |
| U22(X1, mark(X2), X3) | → | U22(X1, X2, X3) | | U22(X1, X2, mark(X3)) | → | U22(X1, X2, X3) |
| U22(active(X1), X2, X3) | → | U22(X1, X2, X3) | | U22(X1, active(X2), X3) | → | U22(X1, X2, X3) |
| U22(X1, X2, active(X3)) | → | U22(X1, X2, X3) | | x(mark(X1), X2) | → | x(X1, X2) |
| x(X1, mark(X2)) | → | x(X1, X2) | | x(active(X1), X2) | → | x(X1, X2) |
| x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, tt, active, U11, mark, U12, U21, U22, x
Strategy
Function Precedence
mark = active < plus = mark# = 0 = s = tt = U11 = U12 = active# = U21 = U22 = x
Argument Filtering
plus: all arguments are removed from plus
mark: all arguments are removed from mark
mark#: all arguments are removed from mark#
0: all arguments are removed from 0
s: all arguments are removed from s
tt: all arguments are removed from tt
U11: all arguments are removed from U11
active: all arguments are removed from active
U12: all arguments are removed from U12
active#: collapses to 1
U21: all arguments are removed from U21
U22: all arguments are removed from U22
x: all arguments are removed from x
Status
plus: multiset
mark: multiset
mark#: multiset
0: multiset
s: multiset
tt: multiset
U11: multiset
active: multiset
U12: multiset
U21: multiset
U22: multiset
x: multiset
Usable Rules
| mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3)) | U12(X1, active(X2), X3) → U12(X1, X2, X3) |
| active(x(N, 0)) → mark(0) | U22(X1, active(X2), X3) → U22(X1, X2, X3) |
| mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2))) | U22(X1, X2, active(X3)) → U22(X1, X2, X3) |
| U21(mark(X1), X2, X3) → U21(X1, X2, X3) | U12(X1, X2, mark(X3)) → U12(X1, X2, X3) |
| plus(mark(X1), X2) → plus(X1, X2) | U22(mark(X1), X2, X3) → U22(X1, X2, X3) |
| U11(mark(X1), X2, X3) → U11(X1, X2, X3) | active(U22(tt, M, N)) → mark(plus(x(N, M), N)) |
| plus(X1, active(X2)) → plus(X1, X2) | U12(X1, mark(X2), X3) → U12(X1, X2, X3) |
| active(U11(tt, M, N)) → mark(U12(tt, M, N)) | U11(X1, mark(X2), X3) → U11(X1, X2, X3) |
| U21(X1, mark(X2), X3) → U21(X1, X2, X3) | active(U21(tt, M, N)) → mark(U22(tt, M, N)) |
| plus(active(X1), X2) → plus(X1, X2) | x(mark(X1), X2) → x(X1, X2) |
| mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3)) | U11(X1, X2, mark(X3)) → U11(X1, X2, X3) |
| U22(X1, mark(X2), X3) → U22(X1, X2, X3) | U21(active(X1), X2, X3) → U21(X1, X2, X3) |
| mark(x(X1, X2)) → active(x(mark(X1), mark(X2))) | mark(s(X)) → active(s(mark(X))) |
| U21(X1, X2, mark(X3)) → U21(X1, X2, X3) | U11(X1, active(X2), X3) → U11(X1, X2, X3) |
| U11(X1, X2, active(X3)) → U11(X1, X2, X3) | mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3)) |
| x(active(X1), X2) → x(X1, X2) | active(x(N, s(M))) → mark(U21(tt, M, N)) |
| U22(active(X1), X2, X3) → U22(X1, X2, X3) | x(X1, active(X2)) → x(X1, X2) |
| s(active(X)) → s(X) | mark(0) → active(0) |
| mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3)) | U11(active(X1), X2, X3) → U11(X1, X2, X3) |
| plus(X1, mark(X2)) → plus(X1, X2) | U21(X1, X2, active(X3)) → U21(X1, X2, X3) |
| active(U12(tt, M, N)) → mark(s(plus(N, M))) | mark(tt) → active(tt) |
| U21(X1, active(X2), X3) → U21(X1, X2, X3) | U12(mark(X1), X2, X3) → U12(X1, X2, X3) |
| U22(X1, X2, mark(X3)) → U22(X1, X2, X3) | x(X1, mark(X2)) → x(X1, X2) |
| active(plus(N, 0)) → mark(N) | s(mark(X)) → s(X) |
| U12(active(X1), X2, X3) → U12(X1, X2, X3) | U12(X1, X2, active(X3)) → U12(X1, X2, X3) |
| active(plus(N, s(M))) → mark(U11(tt, M, N)) |
The dependency pairs and usable rules are stronlgy conservative!
Eliminated dependency pairs
The following dependency pairs (at least) can be eliminated according to the given precedence.
Problem 7: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| plus#(X1, mark(X2)) | → | plus#(X1, X2) | | plus#(X1, active(X2)) | → | plus#(X1, X2) |
| plus#(mark(X1), X2) | → | plus#(X1, X2) | | plus#(active(X1), X2) | → | plus#(X1, X2) |
Rewrite Rules
| active(U11(tt, M, N)) | → | mark(U12(tt, M, N)) | | active(U12(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U21(tt, M, N)) | → | mark(U22(tt, M, N)) | | active(U22(tt, M, N)) | → | mark(plus(x(N, M), N)) |
| active(plus(N, 0)) | → | mark(N) | | active(plus(N, s(M))) | → | mark(U11(tt, M, N)) |
| active(x(N, 0)) | → | mark(0) | | active(x(N, s(M))) | → | mark(U21(tt, M, N)) |
| mark(U11(X1, X2, X3)) | → | active(U11(mark(X1), X2, X3)) | | mark(tt) | → | active(tt) |
| mark(U12(X1, X2, X3)) | → | active(U12(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U21(X1, X2, X3)) | → | active(U21(mark(X1), X2, X3)) |
| mark(U22(X1, X2, X3)) | → | active(U22(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| mark(0) | → | active(0) | | U11(mark(X1), X2, X3) | → | U11(X1, X2, X3) |
| U11(X1, mark(X2), X3) | → | U11(X1, X2, X3) | | U11(X1, X2, mark(X3)) | → | U11(X1, X2, X3) |
| U11(active(X1), X2, X3) | → | U11(X1, X2, X3) | | U11(X1, active(X2), X3) | → | U11(X1, X2, X3) |
| U11(X1, X2, active(X3)) | → | U11(X1, X2, X3) | | U12(mark(X1), X2, X3) | → | U12(X1, X2, X3) |
| U12(X1, mark(X2), X3) | → | U12(X1, X2, X3) | | U12(X1, X2, mark(X3)) | → | U12(X1, X2, X3) |
| U12(active(X1), X2, X3) | → | U12(X1, X2, X3) | | U12(X1, active(X2), X3) | → | U12(X1, X2, X3) |
| U12(X1, X2, active(X3)) | → | U12(X1, X2, X3) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | plus(mark(X1), X2) | → | plus(X1, X2) |
| plus(X1, mark(X2)) | → | plus(X1, X2) | | plus(active(X1), X2) | → | plus(X1, X2) |
| plus(X1, active(X2)) | → | plus(X1, X2) | | U21(mark(X1), X2, X3) | → | U21(X1, X2, X3) |
| U21(X1, mark(X2), X3) | → | U21(X1, X2, X3) | | U21(X1, X2, mark(X3)) | → | U21(X1, X2, X3) |
| U21(active(X1), X2, X3) | → | U21(X1, X2, X3) | | U21(X1, active(X2), X3) | → | U21(X1, X2, X3) |
| U21(X1, X2, active(X3)) | → | U21(X1, X2, X3) | | U22(mark(X1), X2, X3) | → | U22(X1, X2, X3) |
| U22(X1, mark(X2), X3) | → | U22(X1, X2, X3) | | U22(X1, X2, mark(X3)) | → | U22(X1, X2, X3) |
| U22(active(X1), X2, X3) | → | U22(X1, X2, X3) | | U22(X1, active(X2), X3) | → | U22(X1, X2, X3) |
| U22(X1, X2, active(X3)) | → | U22(X1, X2, X3) | | x(mark(X1), X2) | → | x(X1, X2) |
| x(X1, mark(X2)) | → | x(X1, X2) | | x(active(X1), X2) | → | x(X1, X2) |
| x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, tt, active, U11, mark, U12, U21, U22, x
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| plus#(mark(X1), X2) | → | plus#(X1, X2) | | plus#(active(X1), X2) | → | plus#(X1, X2) |
Problem 14: ReductionPairSAT
Dependency Pair Problem
Dependency Pairs
| plus#(X1, active(X2)) | → | plus#(X1, X2) | | plus#(X1, mark(X2)) | → | plus#(X1, X2) |
Rewrite Rules
| active(U11(tt, M, N)) | → | mark(U12(tt, M, N)) | | active(U12(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U21(tt, M, N)) | → | mark(U22(tt, M, N)) | | active(U22(tt, M, N)) | → | mark(plus(x(N, M), N)) |
| active(plus(N, 0)) | → | mark(N) | | active(plus(N, s(M))) | → | mark(U11(tt, M, N)) |
| active(x(N, 0)) | → | mark(0) | | active(x(N, s(M))) | → | mark(U21(tt, M, N)) |
| mark(U11(X1, X2, X3)) | → | active(U11(mark(X1), X2, X3)) | | mark(tt) | → | active(tt) |
| mark(U12(X1, X2, X3)) | → | active(U12(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U21(X1, X2, X3)) | → | active(U21(mark(X1), X2, X3)) |
| mark(U22(X1, X2, X3)) | → | active(U22(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| mark(0) | → | active(0) | | U11(mark(X1), X2, X3) | → | U11(X1, X2, X3) |
| U11(X1, mark(X2), X3) | → | U11(X1, X2, X3) | | U11(X1, X2, mark(X3)) | → | U11(X1, X2, X3) |
| U11(active(X1), X2, X3) | → | U11(X1, X2, X3) | | U11(X1, active(X2), X3) | → | U11(X1, X2, X3) |
| U11(X1, X2, active(X3)) | → | U11(X1, X2, X3) | | U12(mark(X1), X2, X3) | → | U12(X1, X2, X3) |
| U12(X1, mark(X2), X3) | → | U12(X1, X2, X3) | | U12(X1, X2, mark(X3)) | → | U12(X1, X2, X3) |
| U12(active(X1), X2, X3) | → | U12(X1, X2, X3) | | U12(X1, active(X2), X3) | → | U12(X1, X2, X3) |
| U12(X1, X2, active(X3)) | → | U12(X1, X2, X3) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | plus(mark(X1), X2) | → | plus(X1, X2) |
| plus(X1, mark(X2)) | → | plus(X1, X2) | | plus(active(X1), X2) | → | plus(X1, X2) |
| plus(X1, active(X2)) | → | plus(X1, X2) | | U21(mark(X1), X2, X3) | → | U21(X1, X2, X3) |
| U21(X1, mark(X2), X3) | → | U21(X1, X2, X3) | | U21(X1, X2, mark(X3)) | → | U21(X1, X2, X3) |
| U21(active(X1), X2, X3) | → | U21(X1, X2, X3) | | U21(X1, active(X2), X3) | → | U21(X1, X2, X3) |
| U21(X1, X2, active(X3)) | → | U21(X1, X2, X3) | | U22(mark(X1), X2, X3) | → | U22(X1, X2, X3) |
| U22(X1, mark(X2), X3) | → | U22(X1, X2, X3) | | U22(X1, X2, mark(X3)) | → | U22(X1, X2, X3) |
| U22(active(X1), X2, X3) | → | U22(X1, X2, X3) | | U22(X1, active(X2), X3) | → | U22(X1, X2, X3) |
| U22(X1, X2, active(X3)) | → | U22(X1, X2, X3) | | x(mark(X1), X2) | → | x(X1, X2) |
| x(X1, mark(X2)) | → | x(X1, X2) | | x(active(X1), X2) | → | x(X1, X2) |
| x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, tt, U11, active, U12, mark, U21, x, U22
Strategy
Function Precedence
mark < active < plus = 0 = s = tt = U11 = plus# = U12 = U21 = x = U22
Argument Filtering
plus: collapses to 1
0: all arguments are removed from 0
s: all arguments are removed from s
tt: all arguments are removed from tt
U11: all arguments are removed from U11
active: 1
plus#: collapses to 2
mark: collapses to 1
U12: all arguments are removed from U12
U21: all arguments are removed from U21
x: all arguments are removed from x
U22: all arguments are removed from U22
Status
0: multiset
s: multiset
tt: multiset
U11: multiset
active: multiset
U12: multiset
U21: multiset
x: multiset
U22: multiset
Usable Rules
There are no usable rules.
The dependency pairs and usable rules are stronlgy conservative!
Eliminated dependency pairs
The following dependency pairs (at least) can be eliminated according to the given precedence.
| plus#(X1, active(X2)) → plus#(X1, X2) |
Problem 8: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| s#(mark(X)) | → | s#(X) | | s#(active(X)) | → | s#(X) |
Rewrite Rules
| active(U11(tt, M, N)) | → | mark(U12(tt, M, N)) | | active(U12(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U21(tt, M, N)) | → | mark(U22(tt, M, N)) | | active(U22(tt, M, N)) | → | mark(plus(x(N, M), N)) |
| active(plus(N, 0)) | → | mark(N) | | active(plus(N, s(M))) | → | mark(U11(tt, M, N)) |
| active(x(N, 0)) | → | mark(0) | | active(x(N, s(M))) | → | mark(U21(tt, M, N)) |
| mark(U11(X1, X2, X3)) | → | active(U11(mark(X1), X2, X3)) | | mark(tt) | → | active(tt) |
| mark(U12(X1, X2, X3)) | → | active(U12(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U21(X1, X2, X3)) | → | active(U21(mark(X1), X2, X3)) |
| mark(U22(X1, X2, X3)) | → | active(U22(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| mark(0) | → | active(0) | | U11(mark(X1), X2, X3) | → | U11(X1, X2, X3) |
| U11(X1, mark(X2), X3) | → | U11(X1, X2, X3) | | U11(X1, X2, mark(X3)) | → | U11(X1, X2, X3) |
| U11(active(X1), X2, X3) | → | U11(X1, X2, X3) | | U11(X1, active(X2), X3) | → | U11(X1, X2, X3) |
| U11(X1, X2, active(X3)) | → | U11(X1, X2, X3) | | U12(mark(X1), X2, X3) | → | U12(X1, X2, X3) |
| U12(X1, mark(X2), X3) | → | U12(X1, X2, X3) | | U12(X1, X2, mark(X3)) | → | U12(X1, X2, X3) |
| U12(active(X1), X2, X3) | → | U12(X1, X2, X3) | | U12(X1, active(X2), X3) | → | U12(X1, X2, X3) |
| U12(X1, X2, active(X3)) | → | U12(X1, X2, X3) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | plus(mark(X1), X2) | → | plus(X1, X2) |
| plus(X1, mark(X2)) | → | plus(X1, X2) | | plus(active(X1), X2) | → | plus(X1, X2) |
| plus(X1, active(X2)) | → | plus(X1, X2) | | U21(mark(X1), X2, X3) | → | U21(X1, X2, X3) |
| U21(X1, mark(X2), X3) | → | U21(X1, X2, X3) | | U21(X1, X2, mark(X3)) | → | U21(X1, X2, X3) |
| U21(active(X1), X2, X3) | → | U21(X1, X2, X3) | | U21(X1, active(X2), X3) | → | U21(X1, X2, X3) |
| U21(X1, X2, active(X3)) | → | U21(X1, X2, X3) | | U22(mark(X1), X2, X3) | → | U22(X1, X2, X3) |
| U22(X1, mark(X2), X3) | → | U22(X1, X2, X3) | | U22(X1, X2, mark(X3)) | → | U22(X1, X2, X3) |
| U22(active(X1), X2, X3) | → | U22(X1, X2, X3) | | U22(X1, active(X2), X3) | → | U22(X1, X2, X3) |
| U22(X1, X2, active(X3)) | → | U22(X1, X2, X3) | | x(mark(X1), X2) | → | x(X1, X2) |
| x(X1, mark(X2)) | → | x(X1, X2) | | x(active(X1), X2) | → | x(X1, X2) |
| x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, tt, active, U11, mark, U12, U21, U22, x
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| s#(mark(X)) | → | s#(X) | | s#(active(X)) | → | s#(X) |
Problem 9: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U11#(X1, X2, mark(X3)) | → | U11#(X1, X2, X3) | | U11#(X1, active(X2), X3) | → | U11#(X1, X2, X3) |
| U11#(active(X1), X2, X3) | → | U11#(X1, X2, X3) | | U11#(X1, mark(X2), X3) | → | U11#(X1, X2, X3) |
| U11#(mark(X1), X2, X3) | → | U11#(X1, X2, X3) | | U11#(X1, X2, active(X3)) | → | U11#(X1, X2, X3) |
Rewrite Rules
| active(U11(tt, M, N)) | → | mark(U12(tt, M, N)) | | active(U12(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U21(tt, M, N)) | → | mark(U22(tt, M, N)) | | active(U22(tt, M, N)) | → | mark(plus(x(N, M), N)) |
| active(plus(N, 0)) | → | mark(N) | | active(plus(N, s(M))) | → | mark(U11(tt, M, N)) |
| active(x(N, 0)) | → | mark(0) | | active(x(N, s(M))) | → | mark(U21(tt, M, N)) |
| mark(U11(X1, X2, X3)) | → | active(U11(mark(X1), X2, X3)) | | mark(tt) | → | active(tt) |
| mark(U12(X1, X2, X3)) | → | active(U12(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U21(X1, X2, X3)) | → | active(U21(mark(X1), X2, X3)) |
| mark(U22(X1, X2, X3)) | → | active(U22(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| mark(0) | → | active(0) | | U11(mark(X1), X2, X3) | → | U11(X1, X2, X3) |
| U11(X1, mark(X2), X3) | → | U11(X1, X2, X3) | | U11(X1, X2, mark(X3)) | → | U11(X1, X2, X3) |
| U11(active(X1), X2, X3) | → | U11(X1, X2, X3) | | U11(X1, active(X2), X3) | → | U11(X1, X2, X3) |
| U11(X1, X2, active(X3)) | → | U11(X1, X2, X3) | | U12(mark(X1), X2, X3) | → | U12(X1, X2, X3) |
| U12(X1, mark(X2), X3) | → | U12(X1, X2, X3) | | U12(X1, X2, mark(X3)) | → | U12(X1, X2, X3) |
| U12(active(X1), X2, X3) | → | U12(X1, X2, X3) | | U12(X1, active(X2), X3) | → | U12(X1, X2, X3) |
| U12(X1, X2, active(X3)) | → | U12(X1, X2, X3) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | plus(mark(X1), X2) | → | plus(X1, X2) |
| plus(X1, mark(X2)) | → | plus(X1, X2) | | plus(active(X1), X2) | → | plus(X1, X2) |
| plus(X1, active(X2)) | → | plus(X1, X2) | | U21(mark(X1), X2, X3) | → | U21(X1, X2, X3) |
| U21(X1, mark(X2), X3) | → | U21(X1, X2, X3) | | U21(X1, X2, mark(X3)) | → | U21(X1, X2, X3) |
| U21(active(X1), X2, X3) | → | U21(X1, X2, X3) | | U21(X1, active(X2), X3) | → | U21(X1, X2, X3) |
| U21(X1, X2, active(X3)) | → | U21(X1, X2, X3) | | U22(mark(X1), X2, X3) | → | U22(X1, X2, X3) |
| U22(X1, mark(X2), X3) | → | U22(X1, X2, X3) | | U22(X1, X2, mark(X3)) | → | U22(X1, X2, X3) |
| U22(active(X1), X2, X3) | → | U22(X1, X2, X3) | | U22(X1, active(X2), X3) | → | U22(X1, X2, X3) |
| U22(X1, X2, active(X3)) | → | U22(X1, X2, X3) | | x(mark(X1), X2) | → | x(X1, X2) |
| x(X1, mark(X2)) | → | x(X1, X2) | | x(active(X1), X2) | → | x(X1, X2) |
| x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, tt, active, U11, mark, U12, U21, U22, x
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U11#(active(X1), X2, X3) | → | U11#(X1, X2, X3) | | U11#(mark(X1), X2, X3) | → | U11#(X1, X2, X3) |
Problem 15: ReductionPairSAT
Dependency Pair Problem
Dependency Pairs
| U11#(X1, X2, mark(X3)) | → | U11#(X1, X2, X3) | | U11#(X1, active(X2), X3) | → | U11#(X1, X2, X3) |
| U11#(X1, mark(X2), X3) | → | U11#(X1, X2, X3) | | U11#(X1, X2, active(X3)) | → | U11#(X1, X2, X3) |
Rewrite Rules
| active(U11(tt, M, N)) | → | mark(U12(tt, M, N)) | | active(U12(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(U21(tt, M, N)) | → | mark(U22(tt, M, N)) | | active(U22(tt, M, N)) | → | mark(plus(x(N, M), N)) |
| active(plus(N, 0)) | → | mark(N) | | active(plus(N, s(M))) | → | mark(U11(tt, M, N)) |
| active(x(N, 0)) | → | mark(0) | | active(x(N, s(M))) | → | mark(U21(tt, M, N)) |
| mark(U11(X1, X2, X3)) | → | active(U11(mark(X1), X2, X3)) | | mark(tt) | → | active(tt) |
| mark(U12(X1, X2, X3)) | → | active(U12(mark(X1), X2, X3)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | | mark(U21(X1, X2, X3)) | → | active(U21(mark(X1), X2, X3)) |
| mark(U22(X1, X2, X3)) | → | active(U22(mark(X1), X2, X3)) | | mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) |
| mark(0) | → | active(0) | | U11(mark(X1), X2, X3) | → | U11(X1, X2, X3) |
| U11(X1, mark(X2), X3) | → | U11(X1, X2, X3) | | U11(X1, X2, mark(X3)) | → | U11(X1, X2, X3) |
| U11(active(X1), X2, X3) | → | U11(X1, X2, X3) | | U11(X1, active(X2), X3) | → | U11(X1, X2, X3) |
| U11(X1, X2, active(X3)) | → | U11(X1, X2, X3) | | U12(mark(X1), X2, X3) | → | U12(X1, X2, X3) |
| U12(X1, mark(X2), X3) | → | U12(X1, X2, X3) | | U12(X1, X2, mark(X3)) | → | U12(X1, X2, X3) |
| U12(active(X1), X2, X3) | → | U12(X1, X2, X3) | | U12(X1, active(X2), X3) | → | U12(X1, X2, X3) |
| U12(X1, X2, active(X3)) | → | U12(X1, X2, X3) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | plus(mark(X1), X2) | → | plus(X1, X2) |
| plus(X1, mark(X2)) | → | plus(X1, X2) | | plus(active(X1), X2) | → | plus(X1, X2) |
| plus(X1, active(X2)) | → | plus(X1, X2) | | U21(mark(X1), X2, X3) | → | U21(X1, X2, X3) |
| U21(X1, mark(X2), X3) | → | U21(X1, X2, X3) | | U21(X1, X2, mark(X3)) | → | U21(X1, X2, X3) |
| U21(active(X1), X2, X3) | → | U21(X1, X2, X3) | | U21(X1, active(X2), X3) | → | U21(X1, X2, X3) |
| U21(X1, X2, active(X3)) | → | U21(X1, X2, X3) | | U22(mark(X1), X2, X3) | → | U22(X1, X2, X3) |
| U22(X1, mark(X2), X3) | → | U22(X1, X2, X3) | | U22(X1, X2, mark(X3)) | → | U22(X1, X2, X3) |
| U22(active(X1), X2, X3) | → | U22(X1, X2, X3) | | U22(X1, active(X2), X3) | → | U22(X1, X2, X3) |
| U22(X1, X2, active(X3)) | → | U22(X1, X2, X3) | | x(mark(X1), X2) | → | x(X1, X2) |
| x(X1, mark(X2)) | → | x(X1, X2) | | x(active(X1), X2) | → | x(X1, X2) |
| x(X1, active(X2)) | → | x(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, tt, U11, active, U12, mark, U21, x, U22
Strategy
Function Precedence
active = mark < plus = U11# = 0 = s = tt = U11 = U12 = U21 = x = U22
Argument Filtering
plus: 1 2
U11#: collapses to 2
0: all arguments are removed from 0
s: collapses to 1
tt: all arguments are removed from tt
U11: all arguments are removed from U11
active: 1
mark: collapses to 1
U12: 2 3
U21: collapses to 1
x: all arguments are removed from x
U22: collapses to 1
Status
plus: lexicographic with permutation 1 → 2 2 → 1
0: multiset
tt: multiset
U11: multiset
active: multiset
U12: lexicographic with permutation 2 → 2 3 → 1
x: multiset
Usable Rules
There are no usable rules.
The dependency pairs and usable rules are stronlgy conservative!
Eliminated dependency pairs
The following dependency pairs (at least) can be eliminated according to the given precedence.
| U11#(X1, active(X2), X3) → U11#(X1, X2, X3) |