YES
The TRS could be proven terminating. The proof took 16277 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (509ms).
| – Problem 2 was processed with processor SubtermCriterion (2ms).
| | – Problem 7 was processed with processor SubtermCriterion (1ms).
| – Problem 3 was processed with processor SubtermCriterion (2ms).
| | – Problem 8 was processed with processor SubtermCriterion (0ms).
| | | – Problem 10 was processed with processor PolynomialLinearRange4 (36ms).
| – Problem 4 was processed with processor SubtermCriterion (2ms).
| | – Problem 9 was processed with processor SubtermCriterion (0ms).
| | | – Problem 11 was processed with processor PolynomialLinearRange4 (15ms).
| – Problem 5 was processed with processor PolynomialLinearRange4 (186ms).
| | – Problem 12 was processed with processor PolynomialLinearRange4 (181ms).
| | | – Problem 13 was processed with processor PolynomialLinearRange4 (305ms).
| | | | – Problem 14 was processed with processor PolynomialLinearRange4 (152ms).
| | | | | – Problem 15 was processed with processor PolynomialLinearRange4 (119ms).
| | | | | | – Problem 16 was processed with processor PolynomialLinearRange4 (130ms).
| | | | | | | – Problem 17 was processed with processor PolynomialLinearRange4 (136ms).
| | | | | | | | – Problem 18 was processed with processor PolynomialLinearRange4 (104ms).
| | | | | | | | | – Problem 19 was processed with processor PolynomialLinearRange4 (81ms).
| | | | | | | | | | – Problem 20 was processed with processor DependencyGraph (0ms).
| – Problem 6 was processed with processor SubtermCriterion (1ms).
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| active#(U11(tt, M, N)) | → | mark#(U12(tt, M, N)) | | mark#(tt) | → | active#(tt) |
| U12#(X1, mark(X2), X3) | → | U12#(X1, X2, X3) | | active#(plus(N, s(M))) | → | mark#(U11(tt, M, N)) |
| mark#(U11(X1, X2, X3)) | → | U11#(mark(X1), X2, X3) | | mark#(s(X)) | → | s#(mark(X)) |
| active#(U12(tt, M, N)) | → | mark#(s(plus(N, M))) | | mark#(U11(X1, X2, X3)) | → | active#(U11(mark(X1), X2, X3)) |
| U11#(X1, X2, mark(X3)) | → | U11#(X1, X2, X3) | | mark#(plus(X1, X2)) | → | mark#(X2) |
| U12#(active(X1), X2, X3) | → | U12#(X1, X2, X3) | | U11#(active(X1), X2, X3) | → | U11#(X1, X2, X3) |
| plus#(X1, mark(X2)) | → | plus#(X1, 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))) | | U12#(X1, X2, active(X3)) | → | U12#(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) | | active#(U12(tt, M, N)) | → | plus#(N, M) |
| U11#(mark(X1), X2, X3) | → | U11#(X1, X2, X3) | | mark#(plus(X1, X2)) | → | plus#(mark(X1), mark(X2)) |
| mark#(plus(X1, X2)) | → | mark#(X1) | | mark#(U12(X1, X2, X3)) | → | U12#(mark(X1), X2, X3) |
| s#(mark(X)) | → | s#(X) | | active#(U11(tt, M, N)) | → | U12#(tt, M, N) |
| U12#(X1, active(X2), X3) | → | U12#(X1, X2, X3) | | active#(U12(tt, M, N)) | → | s#(plus(N, M)) |
| U12#(X1, X2, mark(X3)) | → | U12#(X1, X2, X3) | | plus#(X1, active(X2)) | → | plus#(X1, X2) |
| U11#(X1, mark(X2), X3) | → | U11#(X1, X2, X3) | | active#(plus(N, s(M))) | → | U11#(tt, M, N) |
| s#(active(X)) | → | s#(X) | | U11#(X1, X2, active(X3)) | → | U11#(X1, X2, X3) |
| plus#(mark(X1), X2) | → | plus#(X1, X2) | | mark#(U11(X1, X2, X3)) | → | mark#(X1) |
| 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(plus(N, 0)) | → | mark(N) | | active(plus(N, s(M))) | → | mark(U11(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(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) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, tt, active, U11, mark, U12
Strategy
The following SCCs where found
| mark#(plus(X1, X2)) → active#(plus(mark(X1), mark(X2))) | mark#(s(X)) → active#(s(mark(X))) |
| active#(U11(tt, M, N)) → mark#(U12(tt, M, N)) | active#(plus(N, 0)) → mark#(N) |
| active#(plus(N, s(M))) → mark#(U11(tt, M, N)) | active#(U12(tt, M, N)) → mark#(s(plus(N, M))) |
| mark#(plus(X1, X2)) → mark#(X1) | mark#(U11(X1, X2, X3)) → active#(U11(mark(X1), X2, X3)) |
| 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#(U11(X1, X2, X3)) → mark#(X1) |
| s#(mark(X)) → s#(X) | s#(active(X)) → s#(X) |
| 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#(active(X1), X2, X3) → U12#(X1, X2, X3) |
| U12#(X1, X2, mark(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) |
Problem 2: 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(plus(N, 0)) | → | mark(N) | | active(plus(N, s(M))) | → | mark(U11(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(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) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, tt, active, U11, mark, U12
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 7: SubtermCriterion
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(plus(N, 0)) | → | mark(N) | | active(plus(N, s(M))) | → | mark(U11(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(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) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, tt, U11, active, U12, mark
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| plus#(X1, mark(X2)) | → | plus#(X1, X2) | | plus#(X1, active(X2)) | → | plus#(X1, X2) |
Problem 3: 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(plus(N, 0)) | → | mark(N) | | active(plus(N, s(M))) | → | mark(U11(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(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) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, tt, active, U11, mark, U12
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 8: 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#(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(plus(N, 0)) | → | mark(N) | | active(plus(N, s(M))) | → | mark(U11(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(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) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, tt, U11, active, U12, mark
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U11#(X1, active(X2), X3) | → | U11#(X1, X2, X3) | | U11#(X1, mark(X2), X3) | → | U11#(X1, X2, X3) |
Problem 10: PolynomialLinearRange4
Dependency Pair Problem
Dependency Pairs
| U11#(X1, X2, mark(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(plus(N, 0)) | → | mark(N) | | active(plus(N, s(M))) | → | mark(U11(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(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) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, tt, active, U11, mark, U12
Strategy
Polynomial Interpretation
- 0: 0
- U11(x,y,z): 0
- U11#(x,y,z): z + 1
- U12(x,y,z): 0
- active(x): 2x + 1
- mark(x): 2x + 1
- plus(x,y): 0
- s(x): 0
- tt: 0
There are no usable rules
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| U11#(X1, X2, mark(X3)) | → | U11#(X1, X2, X3) | | U11#(X1, X2, active(X3)) | → | U11#(X1, X2, X3) |
Problem 4: 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#(active(X1), 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(plus(N, 0)) | → | mark(N) | | active(plus(N, s(M))) | → | mark(U11(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(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) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, tt, active, U11, mark, U12
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U12#(mark(X1), X2, X3) | → | U12#(X1, X2, X3) | | U12#(active(X1), X2, X3) | → | U12#(X1, X2, X3) |
Problem 9: 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#(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(plus(N, 0)) | → | mark(N) | | active(plus(N, s(M))) | → | mark(U11(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(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) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, tt, U11, active, U12, mark
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U12#(X1, active(X2), X3) | → | U12#(X1, X2, X3) | | U12#(X1, mark(X2), X3) | → | U12#(X1, X2, X3) |
Problem 11: PolynomialLinearRange4
Dependency Pair Problem
Dependency Pairs
| U12#(X1, X2, active(X3)) | → | U12#(X1, X2, X3) | | U12#(X1, X2, mark(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(plus(N, 0)) | → | mark(N) | | active(plus(N, s(M))) | → | mark(U11(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(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) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, tt, active, U11, mark, U12
Strategy
Polynomial Interpretation
- 0: 0
- U11(x,y,z): 0
- U12(x,y,z): 0
- U12#(x,y,z): z + 1
- active(x): 2x + 1
- mark(x): 2x + 1
- plus(x,y): 0
- s(x): 0
- tt: 0
There are no usable rules
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| U12#(X1, X2, active(X3)) | → | U12#(X1, X2, X3) | | U12#(X1, X2, mark(X3)) | → | U12#(X1, X2, X3) |
Problem 5: PolynomialLinearRange4
Dependency Pair Problem
Dependency Pairs
| mark#(plus(X1, X2)) | → | active#(plus(mark(X1), mark(X2))) | | mark#(s(X)) | → | active#(s(mark(X))) |
| active#(U11(tt, M, N)) | → | mark#(U12(tt, M, N)) | | active#(plus(N, 0)) | → | mark#(N) |
| active#(plus(N, s(M))) | → | mark#(U11(tt, M, N)) | | active#(U12(tt, M, N)) | → | mark#(s(plus(N, M))) |
| mark#(plus(X1, X2)) | → | mark#(X1) | | mark#(U11(X1, X2, X3)) | → | active#(U11(mark(X1), X2, X3)) |
| 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#(U11(X1, X2, X3)) | → | 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(plus(N, 0)) | → | mark(N) | | active(plus(N, s(M))) | → | mark(U11(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(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) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, tt, active, U11, mark, U12
Strategy
Polynomial Interpretation
- 0: 3
- U11(x,y,z): 1
- U12(x,y,z): 1
- active(x): 0
- active#(x): x
- mark(x): 0
- mark#(x): 1
- plus(x,y): 1
- s(x): 0
- tt: 2
Standard Usable rules
| mark(U12(X1, X2, X3)) | → | active(U12(mark(X1), X2, X3)) | | U12(X1, active(X2), X3) | → | U12(X1, X2, X3) |
| mark(s(X)) | → | active(s(mark(X))) | | mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) |
| U11(X1, active(X2), X3) | → | U11(X1, X2, X3) | | U12(X1, X2, mark(X3)) | → | U12(X1, X2, X3) |
| U11(X1, X2, active(X3)) | → | U11(X1, X2, X3) | | plus(mark(X1), X2) | → | plus(X1, X2) |
| mark(U11(X1, X2, X3)) | → | active(U11(mark(X1), X2, X3)) | | U11(mark(X1), X2, X3) | → | U11(X1, X2, X3) |
| mark(0) | → | active(0) | | s(active(X)) | → | s(X) |
| plus(X1, active(X2)) | → | plus(X1, X2) | | U11(active(X1), X2, X3) | → | U11(X1, X2, X3) |
| U12(X1, mark(X2), X3) | → | U12(X1, X2, X3) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| active(U11(tt, M, N)) | → | mark(U12(tt, M, N)) | | active(U12(tt, M, N)) | → | mark(s(plus(N, M))) |
| mark(tt) | → | active(tt) | | U11(X1, mark(X2), X3) | → | U11(X1, X2, X3) |
| U12(mark(X1), X2, X3) | → | U12(X1, X2, X3) | | plus(active(X1), X2) | → | plus(X1, X2) |
| U11(X1, X2, mark(X3)) | → | U11(X1, X2, X3) | | 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 following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| mark#(s(X)) | → | active#(s(mark(X))) |
Problem 12: PolynomialLinearRange4
Dependency Pair Problem
Dependency Pairs
| mark#(U11(X1, X2, X3)) | → | active#(U11(mark(X1), X2, X3)) | | mark#(plus(X1, X2)) | → | active#(plus(mark(X1), mark(X2))) |
| mark#(plus(X1, X2)) | → | mark#(X2) | | active#(U11(tt, M, N)) | → | mark#(U12(tt, M, N)) |
| active#(plus(N, 0)) | → | mark#(N) | | active#(plus(N, s(M))) | → | mark#(U11(tt, M, N)) |
| mark#(U12(X1, X2, X3)) | → | active#(U12(mark(X1), X2, X3)) | | mark#(s(X)) | → | mark#(X) |
| mark#(U12(X1, X2, X3)) | → | mark#(X1) | | mark#(U11(X1, X2, X3)) | → | mark#(X1) |
| active#(U12(tt, M, N)) | → | mark#(s(plus(N, M))) | | mark#(plus(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(plus(N, 0)) | → | mark(N) | | active(plus(N, s(M))) | → | mark(U11(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(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) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, tt, U11, active, U12, mark
Strategy
Polynomial Interpretation
- 0: 1
- U11(x,y,z): 3z + y + x
- U12(x,y,z): 3z + y + 2x
- active(x): x
- active#(x): 2x
- mark(x): x
- mark#(x): 2x
- plus(x,y): y + 3x
- s(x): x
- tt: 0
Standard Usable rules
| mark(U12(X1, X2, X3)) | → | active(U12(mark(X1), X2, X3)) | | U12(X1, active(X2), X3) | → | U12(X1, X2, X3) |
| mark(s(X)) | → | active(s(mark(X))) | | mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) |
| U11(X1, active(X2), X3) | → | U11(X1, X2, X3) | | U12(X1, X2, mark(X3)) | → | U12(X1, X2, X3) |
| U11(X1, X2, active(X3)) | → | U11(X1, X2, X3) | | plus(mark(X1), X2) | → | plus(X1, X2) |
| mark(U11(X1, X2, X3)) | → | active(U11(mark(X1), X2, X3)) | | U11(mark(X1), X2, X3) | → | U11(X1, X2, X3) |
| mark(0) | → | active(0) | | s(active(X)) | → | s(X) |
| plus(X1, active(X2)) | → | plus(X1, X2) | | U11(active(X1), X2, X3) | → | U11(X1, X2, X3) |
| U12(X1, mark(X2), X3) | → | U12(X1, X2, X3) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| active(U11(tt, M, N)) | → | mark(U12(tt, M, N)) | | active(U12(tt, M, N)) | → | mark(s(plus(N, M))) |
| mark(tt) | → | active(tt) | | U11(X1, mark(X2), X3) | → | U11(X1, X2, X3) |
| U12(mark(X1), X2, X3) | → | U12(X1, X2, X3) | | plus(active(X1), X2) | → | plus(X1, X2) |
| U11(X1, X2, mark(X3)) | → | U11(X1, X2, X3) | | 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 following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| active#(plus(N, 0)) | → | mark#(N) |
Problem 13: PolynomialLinearRange4
Dependency Pair Problem
Dependency Pairs
| mark#(U11(X1, X2, X3)) | → | active#(U11(mark(X1), X2, X3)) | | mark#(plus(X1, X2)) | → | active#(plus(mark(X1), mark(X2))) |
| mark#(plus(X1, X2)) | → | mark#(X2) | | active#(U11(tt, M, N)) | → | mark#(U12(tt, M, N)) |
| active#(plus(N, s(M))) | → | mark#(U11(tt, M, N)) | | mark#(U12(X1, X2, X3)) | → | active#(U12(mark(X1), X2, X3)) |
| mark#(s(X)) | → | mark#(X) | | mark#(U12(X1, X2, X3)) | → | mark#(X1) |
| active#(U12(tt, M, N)) | → | mark#(s(plus(N, M))) | | mark#(U11(X1, X2, X3)) | → | mark#(X1) |
| mark#(plus(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(plus(N, 0)) | → | mark(N) | | active(plus(N, s(M))) | → | mark(U11(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(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) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, tt, active, U11, mark, U12
Strategy
Polynomial Interpretation
- 0: 0
- U11(x,y,z): z + 2y + 2x + 2
- U12(x,y,z): z + 2y + 2x + 2
- active(x): x
- active#(x): x
- mark(x): x
- mark#(x): x
- plus(x,y): 2y + x + 2
- s(x): x
- tt: 0
Standard Usable rules
| mark(U12(X1, X2, X3)) | → | active(U12(mark(X1), X2, X3)) | | U12(X1, active(X2), X3) | → | U12(X1, X2, X3) |
| mark(s(X)) | → | active(s(mark(X))) | | mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) |
| U11(X1, active(X2), X3) | → | U11(X1, X2, X3) | | U12(X1, X2, mark(X3)) | → | U12(X1, X2, X3) |
| U11(X1, X2, active(X3)) | → | U11(X1, X2, X3) | | plus(mark(X1), X2) | → | plus(X1, X2) |
| mark(U11(X1, X2, X3)) | → | active(U11(mark(X1), X2, X3)) | | U11(mark(X1), X2, X3) | → | U11(X1, X2, X3) |
| mark(0) | → | active(0) | | s(active(X)) | → | s(X) |
| plus(X1, active(X2)) | → | plus(X1, X2) | | U11(active(X1), X2, X3) | → | U11(X1, X2, X3) |
| U12(X1, mark(X2), X3) | → | U12(X1, X2, X3) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| active(U11(tt, M, N)) | → | mark(U12(tt, M, N)) | | active(U12(tt, M, N)) | → | mark(s(plus(N, M))) |
| mark(tt) | → | active(tt) | | U11(X1, mark(X2), X3) | → | U11(X1, X2, X3) |
| U12(mark(X1), X2, X3) | → | U12(X1, X2, X3) | | plus(active(X1), X2) | → | plus(X1, X2) |
| U11(X1, X2, mark(X3)) | → | U11(X1, X2, X3) | | 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 following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| mark#(plus(X1, X2)) | → | mark#(X2) | | mark#(U12(X1, X2, X3)) | → | mark#(X1) |
| mark#(U11(X1, X2, X3)) | → | mark#(X1) | | mark#(plus(X1, X2)) | → | mark#(X1) |
Problem 14: PolynomialLinearRange4
Dependency Pair Problem
Dependency Pairs
| mark#(U11(X1, X2, X3)) | → | active#(U11(mark(X1), X2, X3)) | | mark#(plus(X1, X2)) | → | active#(plus(mark(X1), mark(X2))) |
| active#(U11(tt, M, N)) | → | mark#(U12(tt, M, N)) | | active#(plus(N, s(M))) | → | mark#(U11(tt, M, N)) |
| mark#(U12(X1, X2, X3)) | → | active#(U12(mark(X1), X2, X3)) | | mark#(s(X)) | → | mark#(X) |
| active#(U12(tt, M, N)) | → | mark#(s(plus(N, M))) |
Rewrite Rules
| active(U11(tt, M, N)) | → | mark(U12(tt, M, N)) | | active(U12(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(plus(N, 0)) | → | mark(N) | | active(plus(N, s(M))) | → | mark(U11(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(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) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, tt, U11, active, U12, mark
Strategy
Polynomial Interpretation
- 0: 0
- U11(x,y,z): 2z + y + 2x
- U12(x,y,z): 2z + y + 2
- active(x): x
- active#(x): x + 1
- mark(x): x
- mark#(x): x + 1
- plus(x,y): y + 2x
- s(x): x + 2
- tt: 1
Standard Usable rules
| mark(U12(X1, X2, X3)) | → | active(U12(mark(X1), X2, X3)) | | U12(X1, active(X2), X3) | → | U12(X1, X2, X3) |
| mark(s(X)) | → | active(s(mark(X))) | | mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) |
| U11(X1, active(X2), X3) | → | U11(X1, X2, X3) | | U12(X1, X2, mark(X3)) | → | U12(X1, X2, X3) |
| U11(X1, X2, active(X3)) | → | U11(X1, X2, X3) | | plus(mark(X1), X2) | → | plus(X1, X2) |
| mark(U11(X1, X2, X3)) | → | active(U11(mark(X1), X2, X3)) | | U11(mark(X1), X2, X3) | → | U11(X1, X2, X3) |
| mark(0) | → | active(0) | | s(active(X)) | → | s(X) |
| plus(X1, active(X2)) | → | plus(X1, X2) | | U11(active(X1), X2, X3) | → | U11(X1, X2, X3) |
| U12(X1, mark(X2), X3) | → | U12(X1, X2, X3) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| active(U11(tt, M, N)) | → | mark(U12(tt, M, N)) | | active(U12(tt, M, N)) | → | mark(s(plus(N, M))) |
| mark(tt) | → | active(tt) | | U11(X1, mark(X2), X3) | → | U11(X1, X2, X3) |
| U12(mark(X1), X2, X3) | → | U12(X1, X2, X3) | | plus(active(X1), X2) | → | plus(X1, X2) |
| U11(X1, X2, mark(X3)) | → | U11(X1, X2, X3) | | 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 following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
Problem 15: PolynomialLinearRange4
Dependency Pair Problem
Dependency Pairs
| mark#(U11(X1, X2, X3)) | → | active#(U11(mark(X1), X2, X3)) | | mark#(plus(X1, X2)) | → | active#(plus(mark(X1), mark(X2))) |
| active#(U11(tt, M, N)) | → | mark#(U12(tt, M, N)) | | active#(plus(N, s(M))) | → | mark#(U11(tt, M, N)) |
| mark#(U12(X1, X2, X3)) | → | active#(U12(mark(X1), X2, X3)) | | active#(U12(tt, M, N)) | → | mark#(s(plus(N, M))) |
Rewrite Rules
| active(U11(tt, M, N)) | → | mark(U12(tt, M, N)) | | active(U12(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(plus(N, 0)) | → | mark(N) | | active(plus(N, s(M))) | → | mark(U11(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(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) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, tt, active, U11, mark, U12
Strategy
Polynomial Interpretation
- 0: 0
- U11(x,y,z): z + 2y + x
- U12(x,y,z): z + 2y + 1
- active(x): x
- active#(x): x
- mark(x): x
- mark#(x): x
- plus(x,y): 2y + x
- s(x): x + 1
- tt: 1
Standard Usable rules
| mark(U12(X1, X2, X3)) | → | active(U12(mark(X1), X2, X3)) | | U12(X1, active(X2), X3) | → | U12(X1, X2, X3) |
| mark(s(X)) | → | active(s(mark(X))) | | mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) |
| U11(X1, active(X2), X3) | → | U11(X1, X2, X3) | | U12(X1, X2, mark(X3)) | → | U12(X1, X2, X3) |
| U11(X1, X2, active(X3)) | → | U11(X1, X2, X3) | | plus(mark(X1), X2) | → | plus(X1, X2) |
| mark(U11(X1, X2, X3)) | → | active(U11(mark(X1), X2, X3)) | | U11(mark(X1), X2, X3) | → | U11(X1, X2, X3) |
| mark(0) | → | active(0) | | s(active(X)) | → | s(X) |
| plus(X1, active(X2)) | → | plus(X1, X2) | | U11(active(X1), X2, X3) | → | U11(X1, X2, X3) |
| U12(X1, mark(X2), X3) | → | U12(X1, X2, X3) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| active(U11(tt, M, N)) | → | mark(U12(tt, M, N)) | | active(U12(tt, M, N)) | → | mark(s(plus(N, M))) |
| mark(tt) | → | active(tt) | | U11(X1, mark(X2), X3) | → | U11(X1, X2, X3) |
| U12(mark(X1), X2, X3) | → | U12(X1, X2, X3) | | plus(active(X1), X2) | → | plus(X1, X2) |
| U11(X1, X2, mark(X3)) | → | U11(X1, X2, X3) | | 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 following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| active#(plus(N, s(M))) | → | mark#(U11(tt, M, N)) |
Problem 16: PolynomialLinearRange4
Dependency Pair Problem
Dependency Pairs
| mark#(U11(X1, X2, X3)) | → | active#(U11(mark(X1), X2, X3)) | | mark#(plus(X1, X2)) | → | active#(plus(mark(X1), mark(X2))) |
| active#(U11(tt, M, N)) | → | mark#(U12(tt, M, N)) | | mark#(U12(X1, X2, X3)) | → | active#(U12(mark(X1), X2, X3)) |
| active#(U12(tt, M, N)) | → | mark#(s(plus(N, M))) |
Rewrite Rules
| active(U11(tt, M, N)) | → | mark(U12(tt, M, N)) | | active(U12(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(plus(N, 0)) | → | mark(N) | | active(plus(N, s(M))) | → | mark(U11(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(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) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, tt, U11, active, U12, mark
Strategy
Polynomial Interpretation
- 0: 1
- U11(x,y,z): z + 1
- U12(x,y,z): z + x
- active(x): x
- active#(x): 2x
- mark(x): x
- mark#(x): 2x
- plus(x,y): x + 1
- s(x): 0
- tt: 1
Standard Usable rules
| mark(U12(X1, X2, X3)) | → | active(U12(mark(X1), X2, X3)) | | U12(X1, active(X2), X3) | → | U12(X1, X2, X3) |
| mark(s(X)) | → | active(s(mark(X))) | | mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) |
| U11(X1, active(X2), X3) | → | U11(X1, X2, X3) | | U12(X1, X2, mark(X3)) | → | U12(X1, X2, X3) |
| U11(X1, X2, active(X3)) | → | U11(X1, X2, X3) | | plus(mark(X1), X2) | → | plus(X1, X2) |
| mark(U11(X1, X2, X3)) | → | active(U11(mark(X1), X2, X3)) | | U11(mark(X1), X2, X3) | → | U11(X1, X2, X3) |
| mark(0) | → | active(0) | | s(active(X)) | → | s(X) |
| plus(X1, active(X2)) | → | plus(X1, X2) | | U11(active(X1), X2, X3) | → | U11(X1, X2, X3) |
| U12(X1, mark(X2), X3) | → | U12(X1, X2, X3) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| active(U11(tt, M, N)) | → | mark(U12(tt, M, N)) | | active(U12(tt, M, N)) | → | mark(s(plus(N, M))) |
| mark(tt) | → | active(tt) | | U11(X1, mark(X2), X3) | → | U11(X1, X2, X3) |
| U12(mark(X1), X2, X3) | → | U12(X1, X2, X3) | | plus(active(X1), X2) | → | plus(X1, X2) |
| U11(X1, X2, mark(X3)) | → | U11(X1, X2, X3) | | 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 following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| active#(U12(tt, M, N)) | → | mark#(s(plus(N, M))) |
Problem 17: PolynomialLinearRange4
Dependency Pair Problem
Dependency Pairs
| mark#(U11(X1, X2, X3)) | → | active#(U11(mark(X1), X2, X3)) | | mark#(plus(X1, X2)) | → | active#(plus(mark(X1), mark(X2))) |
| active#(U11(tt, M, N)) | → | mark#(U12(tt, M, N)) | | mark#(U12(X1, X2, X3)) | → | active#(U12(mark(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(plus(N, 0)) | → | mark(N) | | active(plus(N, s(M))) | → | mark(U11(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(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) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, tt, active, U11, mark, U12
Strategy
Polynomial Interpretation
- 0: 1
- U11(x,y,z): 2x
- U12(x,y,z): 0
- active(x): x
- active#(x): 0
- mark(x): x
- mark#(x): x
- plus(x,y): y + x + 1
- s(x): 0
- tt: 0
Standard Usable rules
| mark(U12(X1, X2, X3)) | → | active(U12(mark(X1), X2, X3)) | | U12(X1, active(X2), X3) | → | U12(X1, X2, X3) |
| mark(s(X)) | → | active(s(mark(X))) | | mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) |
| U11(X1, active(X2), X3) | → | U11(X1, X2, X3) | | U12(X1, X2, mark(X3)) | → | U12(X1, X2, X3) |
| U11(X1, X2, active(X3)) | → | U11(X1, X2, X3) | | plus(mark(X1), X2) | → | plus(X1, X2) |
| mark(U11(X1, X2, X3)) | → | active(U11(mark(X1), X2, X3)) | | U11(mark(X1), X2, X3) | → | U11(X1, X2, X3) |
| s(active(X)) | → | s(X) | | mark(0) | → | active(0) |
| plus(X1, active(X2)) | → | plus(X1, X2) | | U11(active(X1), X2, X3) | → | U11(X1, X2, X3) |
| U12(X1, mark(X2), X3) | → | U12(X1, X2, X3) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| active(U12(tt, M, N)) | → | mark(s(plus(N, M))) | | active(U11(tt, M, N)) | → | mark(U12(tt, M, N)) |
| mark(tt) | → | active(tt) | | U11(X1, mark(X2), X3) | → | U11(X1, X2, X3) |
| U12(mark(X1), X2, X3) | → | U12(X1, X2, X3) | | plus(active(X1), X2) | → | plus(X1, X2) |
| U11(X1, X2, mark(X3)) | → | U11(X1, X2, X3) | | 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 following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| mark#(plus(X1, X2)) | → | active#(plus(mark(X1), mark(X2))) |
Problem 18: PolynomialLinearRange4
Dependency Pair Problem
Dependency Pairs
| mark#(U11(X1, X2, X3)) | → | active#(U11(mark(X1), X2, X3)) | | active#(U11(tt, M, N)) | → | mark#(U12(tt, M, N)) |
| mark#(U12(X1, X2, X3)) | → | active#(U12(mark(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(plus(N, 0)) | → | mark(N) | | active(plus(N, s(M))) | → | mark(U11(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(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) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, tt, U11, active, U12, mark
Strategy
Polynomial Interpretation
- 0: 3
- U11(x,y,z): 2
- U12(x,y,z): 0
- active(x): 1
- active#(x): x
- mark(x): 1
- mark#(x): 2
- plus(x,y): 1
- s(x): 0
- tt: 3
Standard Usable rules
| mark(U12(X1, X2, X3)) | → | active(U12(mark(X1), X2, X3)) | | U12(X1, active(X2), X3) | → | U12(X1, X2, X3) |
| mark(s(X)) | → | active(s(mark(X))) | | mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) |
| U11(X1, active(X2), X3) | → | U11(X1, X2, X3) | | U12(X1, X2, mark(X3)) | → | U12(X1, X2, X3) |
| U11(X1, X2, active(X3)) | → | U11(X1, X2, X3) | | plus(mark(X1), X2) | → | plus(X1, X2) |
| mark(U11(X1, X2, X3)) | → | active(U11(mark(X1), X2, X3)) | | U11(mark(X1), X2, X3) | → | U11(X1, X2, X3) |
| s(active(X)) | → | s(X) | | mark(0) | → | active(0) |
| plus(X1, active(X2)) | → | plus(X1, X2) | | U11(active(X1), X2, X3) | → | U11(X1, X2, X3) |
| U12(X1, mark(X2), X3) | → | U12(X1, X2, X3) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| active(U12(tt, M, N)) | → | mark(s(plus(N, M))) | | active(U11(tt, M, N)) | → | mark(U12(tt, M, N)) |
| mark(tt) | → | active(tt) | | U11(X1, mark(X2), X3) | → | U11(X1, X2, X3) |
| U12(mark(X1), X2, X3) | → | U12(X1, X2, X3) | | plus(active(X1), X2) | → | plus(X1, X2) |
| U11(X1, X2, mark(X3)) | → | U11(X1, X2, X3) | | 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 following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| mark#(U12(X1, X2, X3)) | → | active#(U12(mark(X1), X2, X3)) |
Problem 19: PolynomialLinearRange4
Dependency Pair Problem
Dependency Pairs
| mark#(U11(X1, X2, X3)) | → | active#(U11(mark(X1), X2, X3)) | | active#(U11(tt, M, N)) | → | mark#(U12(tt, M, N)) |
Rewrite Rules
| active(U11(tt, M, N)) | → | mark(U12(tt, M, N)) | | active(U12(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(plus(N, 0)) | → | mark(N) | | active(plus(N, s(M))) | → | mark(U11(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(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) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, tt, active, U11, mark, U12
Strategy
Polynomial Interpretation
- 0: 3
- U11(x,y,z): 1
- U12(x,y,z): 0
- active(x): 0
- active#(x): 0
- mark(x): 0
- mark#(x): 2x
- plus(x,y): 1
- s(x): 1
- tt: 3
Standard Usable rules
| mark(U12(X1, X2, X3)) | → | active(U12(mark(X1), X2, X3)) | | U12(X1, active(X2), X3) | → | U12(X1, X2, X3) |
| mark(s(X)) | → | active(s(mark(X))) | | mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) |
| U11(X1, active(X2), X3) | → | U11(X1, X2, X3) | | U12(X1, X2, mark(X3)) | → | U12(X1, X2, X3) |
| U11(X1, X2, active(X3)) | → | U11(X1, X2, X3) | | plus(mark(X1), X2) | → | plus(X1, X2) |
| mark(U11(X1, X2, X3)) | → | active(U11(mark(X1), X2, X3)) | | U11(mark(X1), X2, X3) | → | U11(X1, X2, X3) |
| s(active(X)) | → | s(X) | | mark(0) | → | active(0) |
| plus(X1, active(X2)) | → | plus(X1, X2) | | U11(active(X1), X2, X3) | → | U11(X1, X2, X3) |
| U12(X1, mark(X2), X3) | → | U12(X1, X2, X3) | | plus(X1, mark(X2)) | → | plus(X1, X2) |
| active(U12(tt, M, N)) | → | mark(s(plus(N, M))) | | active(U11(tt, M, N)) | → | mark(U12(tt, M, N)) |
| mark(tt) | → | active(tt) | | U11(X1, mark(X2), X3) | → | U11(X1, X2, X3) |
| U12(mark(X1), X2, X3) | → | U12(X1, X2, X3) | | plus(active(X1), X2) | → | plus(X1, X2) |
| U11(X1, X2, mark(X3)) | → | U11(X1, X2, X3) | | 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 following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| mark#(U11(X1, X2, X3)) | → | active#(U11(mark(X1), X2, X3)) |
Problem 20: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| active#(U11(tt, M, N)) | → | mark#(U12(tt, M, N)) |
Rewrite Rules
| active(U11(tt, M, N)) | → | mark(U12(tt, M, N)) | | active(U12(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(plus(N, 0)) | → | mark(N) | | active(plus(N, s(M))) | → | mark(U11(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(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) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, tt, U11, active, U12, mark
Strategy
There are no SCCs!
Problem 6: 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(plus(N, 0)) | → | mark(N) | | active(plus(N, s(M))) | → | mark(U11(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(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) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, tt, active, U11, mark, U12
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| s#(mark(X)) | → | s#(X) | | s#(active(X)) | → | s#(X) |