TIMEOUT
The TRS could not be proven terminating. The proof attempt took 60001 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (4426ms).
| – Problem 2 remains open; application of the following processors failed [SubtermCriterion (1ms), DependencyGraph (7ms), PolynomialLinearRange4iUR (3333ms), DependencyGraph (6ms), PolynomialLinearRange4iUR (5000ms), DependencyGraph (27ms), PolynomialLinearRange8NegiUR (15000ms), DependencyGraph (7ms), ReductionPairSAT (12899ms), DependencyGraph (5ms), ReductionPairSAT (12773ms), DependencyGraph (5ms), SizeChangePrinciple (timeout)].
| – Problem 3 was processed with processor SubtermCriterion (2ms).
| – Problem 4 was processed with processor SubtermCriterion (3ms).
| | – Problem 12 was processed with processor ReductionPairSAT (90ms).
| – Problem 5 was processed with processor SubtermCriterion (1ms).
| – Problem 6 was processed with processor SubtermCriterion (1ms).
| | – Problem 13 was processed with processor PolynomialLinearRange4iUR (23ms).
| – Problem 7 was processed with processor SubtermCriterion (1ms).
| – Problem 8 was processed with processor SubtermCriterion (1ms).
| – Problem 9 was processed with processor SubtermCriterion (4ms).
| – Problem 10 was processed with processor SubtermCriterion (1ms).
| – Problem 11 was processed with processor SubtermCriterion (1ms).
The following open problems remain:
Open Dependency Pair Problem 2
Dependency Pairs
| top#(mark(X)) | → | top#(proper(X)) | | top#(ok(X)) | → | top#(active(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)) |
| active(U11(X1, X2, X3)) | → | U11(active(X1), X2, X3) | | active(U12(X1, X2, X3)) | → | U12(active(X1), X2, X3) |
| active(s(X)) | → | s(active(X)) | | active(plus(X1, X2)) | → | plus(active(X1), X2) |
| active(plus(X1, X2)) | → | plus(X1, active(X2)) | | active(U21(X1, X2, X3)) | → | U21(active(X1), X2, X3) |
| active(U22(X1, X2, X3)) | → | U22(active(X1), X2, X3) | | active(x(X1, X2)) | → | x(active(X1), X2) |
| active(x(X1, X2)) | → | x(X1, active(X2)) | | U11(mark(X1), X2, X3) | → | mark(U11(X1, X2, X3)) |
| U12(mark(X1), X2, X3) | → | mark(U12(X1, X2, X3)) | | s(mark(X)) | → | mark(s(X)) |
| plus(mark(X1), X2) | → | mark(plus(X1, X2)) | | plus(X1, mark(X2)) | → | mark(plus(X1, X2)) |
| U21(mark(X1), X2, X3) | → | mark(U21(X1, X2, X3)) | | U22(mark(X1), X2, X3) | → | mark(U22(X1, X2, X3)) |
| x(mark(X1), X2) | → | mark(x(X1, X2)) | | x(X1, mark(X2)) | → | mark(x(X1, X2)) |
| proper(U11(X1, X2, X3)) | → | U11(proper(X1), proper(X2), proper(X3)) | | proper(tt) | → | ok(tt) |
| proper(U12(X1, X2, X3)) | → | U12(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(U21(X1, X2, X3)) | → | U21(proper(X1), proper(X2), proper(X3)) |
| proper(U22(X1, X2, X3)) | → | U22(proper(X1), proper(X2), proper(X3)) | | proper(x(X1, X2)) | → | x(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | U11(ok(X1), ok(X2), ok(X3)) | → | ok(U11(X1, X2, X3)) |
| U12(ok(X1), ok(X2), ok(X3)) | → | ok(U12(X1, X2, X3)) | | s(ok(X)) | → | ok(s(X)) |
| plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) | | U21(ok(X1), ok(X2), ok(X3)) | → | ok(U21(X1, X2, X3)) |
| U22(ok(X1), ok(X2), ok(X3)) | → | ok(U22(X1, X2, X3)) | | x(ok(X1), ok(X2)) | → | ok(x(X1, X2)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, 0, s, tt, active, U11, U12, ok, proper, U21, top, U22, x
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| active#(x(N, s(M))) | → | U21#(tt, M, N) | | top#(ok(X)) | → | top#(active(X)) |
| U21#(ok(X1), ok(X2), ok(X3)) | → | U21#(X1, X2, X3) | | proper#(U11(X1, X2, X3)) | → | proper#(X3) |
| U22#(mark(X1), X2, X3) | → | U22#(X1, X2, X3) | | proper#(U22(X1, X2, X3)) | → | U22#(proper(X1), proper(X2), proper(X3)) |
| proper#(U11(X1, X2, X3)) | → | proper#(X2) | | plus#(X1, mark(X2)) | → | plus#(X1, X2) |
| proper#(plus(X1, X2)) | → | proper#(X1) | | x#(ok(X1), ok(X2)) | → | x#(X1, X2) |
| x#(X1, mark(X2)) | → | x#(X1, X2) | | top#(mark(X)) | → | proper#(X) |
| proper#(plus(X1, X2)) | → | plus#(proper(X1), proper(X2)) | | top#(mark(X)) | → | top#(proper(X)) |
| U12#(mark(X1), X2, X3) | → | U12#(X1, X2, X3) | | proper#(U22(X1, X2, X3)) | → | proper#(X3) |
| U21#(mark(X1), X2, X3) | → | U21#(X1, X2, X3) | | U11#(mark(X1), X2, X3) | → | U11#(X1, X2, X3) |
| active#(U21(X1, X2, X3)) | → | U21#(active(X1), X2, X3) | | active#(U22(X1, X2, X3)) | → | U22#(active(X1), X2, X3) |
| proper#(U12(X1, X2, X3)) | → | proper#(X3) | | active#(x(X1, X2)) | → | x#(X1, active(X2)) |
| active#(U21(X1, X2, X3)) | → | active#(X1) | | active#(U11(X1, X2, X3)) | → | active#(X1) |
| active#(U12(X1, X2, X3)) | → | active#(X1) | | proper#(s(X)) | → | proper#(X) |
| active#(plus(X1, X2)) | → | active#(X1) | | U12#(ok(X1), ok(X2), ok(X3)) | → | U12#(X1, X2, X3) |
| active#(U12(tt, M, N)) | → | s#(plus(N, M)) | | U22#(ok(X1), ok(X2), ok(X3)) | → | U22#(X1, X2, X3) |
| active#(plus(X1, X2)) | → | active#(X2) | | U11#(ok(X1), ok(X2), ok(X3)) | → | U11#(X1, X2, X3) |
| plus#(mark(X1), X2) | → | plus#(X1, X2) | | proper#(U12(X1, X2, X3)) | → | proper#(X1) |
| proper#(U21(X1, X2, X3)) | → | proper#(X3) | | proper#(x(X1, X2)) | → | proper#(X1) |
| active#(U11(X1, X2, X3)) | → | U11#(active(X1), X2, X3) | | active#(x(X1, X2)) | → | x#(active(X1), X2) |
| top#(ok(X)) | → | active#(X) | | active#(U12(X1, X2, X3)) | → | U12#(active(X1), X2, X3) |
| x#(mark(X1), X2) | → | x#(X1, X2) | | proper#(U21(X1, X2, X3)) | → | proper#(X1) |
| proper#(U22(X1, X2, X3)) | → | proper#(X1) | | proper#(x(X1, X2)) | → | x#(proper(X1), proper(X2)) |
| proper#(plus(X1, X2)) | → | proper#(X2) | | plus#(ok(X1), ok(X2)) | → | plus#(X1, X2) |
| active#(U21(tt, M, N)) | → | U22#(tt, M, N) | | proper#(U12(X1, X2, X3)) | → | proper#(X2) |
| proper#(U12(X1, X2, X3)) | → | U12#(proper(X1), proper(X2), proper(X3)) | | proper#(U22(X1, X2, X3)) | → | proper#(X2) |
| active#(U22(tt, M, N)) | → | x#(N, M) | | active#(x(X1, X2)) | → | active#(X1) |
| active#(U12(tt, M, N)) | → | plus#(N, M) | | proper#(U21(X1, X2, X3)) | → | U21#(proper(X1), proper(X2), proper(X3)) |
| active#(s(X)) | → | s#(active(X)) | | proper#(x(X1, X2)) | → | proper#(X2) |
| s#(ok(X)) | → | s#(X) | | proper#(U11(X1, X2, X3)) | → | U11#(proper(X1), proper(X2), proper(X3)) |
| proper#(U11(X1, X2, X3)) | → | proper#(X1) | | s#(mark(X)) | → | s#(X) |
| active#(U22(tt, M, N)) | → | plus#(x(N, M), N) | | active#(plus(X1, X2)) | → | plus#(X1, active(X2)) |
| active#(U22(X1, X2, X3)) | → | active#(X1) | | active#(U11(tt, M, N)) | → | U12#(tt, M, N) |
| active#(s(X)) | → | active#(X) | | active#(plus(X1, X2)) | → | plus#(active(X1), X2) |
| proper#(s(X)) | → | s#(proper(X)) | | active#(x(X1, X2)) | → | active#(X2) |
| proper#(U21(X1, X2, X3)) | → | proper#(X2) | | active#(plus(N, s(M))) | → | U11#(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(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)) |
| active(U11(X1, X2, X3)) | → | U11(active(X1), X2, X3) | | active(U12(X1, X2, X3)) | → | U12(active(X1), X2, X3) |
| active(s(X)) | → | s(active(X)) | | active(plus(X1, X2)) | → | plus(active(X1), X2) |
| active(plus(X1, X2)) | → | plus(X1, active(X2)) | | active(U21(X1, X2, X3)) | → | U21(active(X1), X2, X3) |
| active(U22(X1, X2, X3)) | → | U22(active(X1), X2, X3) | | active(x(X1, X2)) | → | x(active(X1), X2) |
| active(x(X1, X2)) | → | x(X1, active(X2)) | | U11(mark(X1), X2, X3) | → | mark(U11(X1, X2, X3)) |
| U12(mark(X1), X2, X3) | → | mark(U12(X1, X2, X3)) | | s(mark(X)) | → | mark(s(X)) |
| plus(mark(X1), X2) | → | mark(plus(X1, X2)) | | plus(X1, mark(X2)) | → | mark(plus(X1, X2)) |
| U21(mark(X1), X2, X3) | → | mark(U21(X1, X2, X3)) | | U22(mark(X1), X2, X3) | → | mark(U22(X1, X2, X3)) |
| x(mark(X1), X2) | → | mark(x(X1, X2)) | | x(X1, mark(X2)) | → | mark(x(X1, X2)) |
| proper(U11(X1, X2, X3)) | → | U11(proper(X1), proper(X2), proper(X3)) | | proper(tt) | → | ok(tt) |
| proper(U12(X1, X2, X3)) | → | U12(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(U21(X1, X2, X3)) | → | U21(proper(X1), proper(X2), proper(X3)) |
| proper(U22(X1, X2, X3)) | → | U22(proper(X1), proper(X2), proper(X3)) | | proper(x(X1, X2)) | → | x(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | U11(ok(X1), ok(X2), ok(X3)) | → | ok(U11(X1, X2, X3)) |
| U12(ok(X1), ok(X2), ok(X3)) | → | ok(U12(X1, X2, X3)) | | s(ok(X)) | → | ok(s(X)) |
| plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) | | U21(ok(X1), ok(X2), ok(X3)) | → | ok(U21(X1, X2, X3)) |
| U22(ok(X1), ok(X2), ok(X3)) | → | ok(U22(X1, X2, X3)) | | x(ok(X1), ok(X2)) | → | ok(x(X1, X2)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, 0, s, tt, active, U11, U12, ok, proper, U21, x, top, U22
Strategy
The following SCCs where found
| plus#(ok(X1), ok(X2)) → plus#(X1, X2) | plus#(X1, mark(X2)) → plus#(X1, X2) |
| plus#(mark(X1), X2) → plus#(X1, X2) |
| proper#(U11(X1, X2, X3)) → proper#(X3) | proper#(U12(X1, X2, X3)) → proper#(X2) |
| proper#(U22(X1, X2, X3)) → proper#(X2) | proper#(U21(X1, X2, X3)) → proper#(X3) |
| proper#(U22(X1, X2, X3)) → proper#(X3) | proper#(x(X1, X2)) → proper#(X1) |
| proper#(x(X1, X2)) → proper#(X2) | proper#(U12(X1, X2, X3)) → proper#(X3) |
| proper#(U11(X1, X2, X3)) → proper#(X1) | proper#(s(X)) → proper#(X) |
| proper#(U11(X1, X2, X3)) → proper#(X2) | proper#(U21(X1, X2, X3)) → proper#(X1) |
| proper#(U22(X1, X2, X3)) → proper#(X1) | proper#(plus(X1, X2)) → proper#(X1) |
| proper#(U21(X1, X2, X3)) → proper#(X2) | proper#(plus(X1, X2)) → proper#(X2) |
| proper#(U12(X1, X2, X3)) → proper#(X1) |
| U21#(ok(X1), ok(X2), ok(X3)) → U21#(X1, X2, X3) | U21#(mark(X1), X2, X3) → U21#(X1, X2, X3) |
| U22#(mark(X1), X2, X3) → U22#(X1, X2, X3) | U22#(ok(X1), ok(X2), ok(X3)) → U22#(X1, X2, X3) |
| x#(mark(X1), X2) → x#(X1, X2) | x#(ok(X1), ok(X2)) → x#(X1, X2) |
| x#(X1, mark(X2)) → x#(X1, X2) |
| active#(U11(X1, X2, X3)) → active#(X1) | active#(U22(X1, X2, X3)) → active#(X1) |
| active#(U12(X1, X2, X3)) → active#(X1) | active#(plus(X1, X2)) → active#(X1) |
| active#(s(X)) → active#(X) | active#(x(X1, X2)) → active#(X1) |
| active#(x(X1, X2)) → active#(X2) | active#(plus(X1, X2)) → active#(X2) |
| active#(U21(X1, X2, X3)) → active#(X1) |
| U11#(mark(X1), X2, X3) → U11#(X1, X2, X3) | U11#(ok(X1), ok(X2), ok(X3)) → U11#(X1, X2, X3) |
| s#(mark(X)) → s#(X) | s#(ok(X)) → s#(X) |
| U12#(ok(X1), ok(X2), ok(X3)) → U12#(X1, X2, X3) | U12#(mark(X1), X2, X3) → U12#(X1, X2, X3) |
| top#(mark(X)) → top#(proper(X)) | top#(ok(X)) → top#(active(X)) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| s#(mark(X)) | → | s#(X) | | s#(ok(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)) |
| active(U11(X1, X2, X3)) | → | U11(active(X1), X2, X3) | | active(U12(X1, X2, X3)) | → | U12(active(X1), X2, X3) |
| active(s(X)) | → | s(active(X)) | | active(plus(X1, X2)) | → | plus(active(X1), X2) |
| active(plus(X1, X2)) | → | plus(X1, active(X2)) | | active(U21(X1, X2, X3)) | → | U21(active(X1), X2, X3) |
| active(U22(X1, X2, X3)) | → | U22(active(X1), X2, X3) | | active(x(X1, X2)) | → | x(active(X1), X2) |
| active(x(X1, X2)) | → | x(X1, active(X2)) | | U11(mark(X1), X2, X3) | → | mark(U11(X1, X2, X3)) |
| U12(mark(X1), X2, X3) | → | mark(U12(X1, X2, X3)) | | s(mark(X)) | → | mark(s(X)) |
| plus(mark(X1), X2) | → | mark(plus(X1, X2)) | | plus(X1, mark(X2)) | → | mark(plus(X1, X2)) |
| U21(mark(X1), X2, X3) | → | mark(U21(X1, X2, X3)) | | U22(mark(X1), X2, X3) | → | mark(U22(X1, X2, X3)) |
| x(mark(X1), X2) | → | mark(x(X1, X2)) | | x(X1, mark(X2)) | → | mark(x(X1, X2)) |
| proper(U11(X1, X2, X3)) | → | U11(proper(X1), proper(X2), proper(X3)) | | proper(tt) | → | ok(tt) |
| proper(U12(X1, X2, X3)) | → | U12(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(U21(X1, X2, X3)) | → | U21(proper(X1), proper(X2), proper(X3)) |
| proper(U22(X1, X2, X3)) | → | U22(proper(X1), proper(X2), proper(X3)) | | proper(x(X1, X2)) | → | x(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | U11(ok(X1), ok(X2), ok(X3)) | → | ok(U11(X1, X2, X3)) |
| U12(ok(X1), ok(X2), ok(X3)) | → | ok(U12(X1, X2, X3)) | | s(ok(X)) | → | ok(s(X)) |
| plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) | | U21(ok(X1), ok(X2), ok(X3)) | → | ok(U21(X1, X2, X3)) |
| U22(ok(X1), ok(X2), ok(X3)) | → | ok(U22(X1, X2, X3)) | | x(ok(X1), ok(X2)) | → | ok(x(X1, X2)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, 0, s, tt, active, U11, U12, ok, proper, U21, x, top, U22
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| s#(mark(X)) | → | s#(X) | | s#(ok(X)) | → | s#(X) |
Problem 4: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| x#(mark(X1), X2) | → | x#(X1, X2) | | x#(ok(X1), ok(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)) |
| active(U11(X1, X2, X3)) | → | U11(active(X1), X2, X3) | | active(U12(X1, X2, X3)) | → | U12(active(X1), X2, X3) |
| active(s(X)) | → | s(active(X)) | | active(plus(X1, X2)) | → | plus(active(X1), X2) |
| active(plus(X1, X2)) | → | plus(X1, active(X2)) | | active(U21(X1, X2, X3)) | → | U21(active(X1), X2, X3) |
| active(U22(X1, X2, X3)) | → | U22(active(X1), X2, X3) | | active(x(X1, X2)) | → | x(active(X1), X2) |
| active(x(X1, X2)) | → | x(X1, active(X2)) | | U11(mark(X1), X2, X3) | → | mark(U11(X1, X2, X3)) |
| U12(mark(X1), X2, X3) | → | mark(U12(X1, X2, X3)) | | s(mark(X)) | → | mark(s(X)) |
| plus(mark(X1), X2) | → | mark(plus(X1, X2)) | | plus(X1, mark(X2)) | → | mark(plus(X1, X2)) |
| U21(mark(X1), X2, X3) | → | mark(U21(X1, X2, X3)) | | U22(mark(X1), X2, X3) | → | mark(U22(X1, X2, X3)) |
| x(mark(X1), X2) | → | mark(x(X1, X2)) | | x(X1, mark(X2)) | → | mark(x(X1, X2)) |
| proper(U11(X1, X2, X3)) | → | U11(proper(X1), proper(X2), proper(X3)) | | proper(tt) | → | ok(tt) |
| proper(U12(X1, X2, X3)) | → | U12(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(U21(X1, X2, X3)) | → | U21(proper(X1), proper(X2), proper(X3)) |
| proper(U22(X1, X2, X3)) | → | U22(proper(X1), proper(X2), proper(X3)) | | proper(x(X1, X2)) | → | x(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | U11(ok(X1), ok(X2), ok(X3)) | → | ok(U11(X1, X2, X3)) |
| U12(ok(X1), ok(X2), ok(X3)) | → | ok(U12(X1, X2, X3)) | | s(ok(X)) | → | ok(s(X)) |
| plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) | | U21(ok(X1), ok(X2), ok(X3)) | → | ok(U21(X1, X2, X3)) |
| U22(ok(X1), ok(X2), ok(X3)) | → | ok(U22(X1, X2, X3)) | | x(ok(X1), ok(X2)) | → | ok(x(X1, X2)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, 0, s, tt, active, U11, U12, ok, proper, U21, x, top, U22
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| x#(mark(X1), X2) | → | x#(X1, X2) | | x#(ok(X1), ok(X2)) | → | x#(X1, X2) |
Problem 12: ReductionPairSAT
Dependency Pair Problem
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)) |
| active(U11(X1, X2, X3)) | → | U11(active(X1), X2, X3) | | active(U12(X1, X2, X3)) | → | U12(active(X1), X2, X3) |
| active(s(X)) | → | s(active(X)) | | active(plus(X1, X2)) | → | plus(active(X1), X2) |
| active(plus(X1, X2)) | → | plus(X1, active(X2)) | | active(U21(X1, X2, X3)) | → | U21(active(X1), X2, X3) |
| active(U22(X1, X2, X3)) | → | U22(active(X1), X2, X3) | | active(x(X1, X2)) | → | x(active(X1), X2) |
| active(x(X1, X2)) | → | x(X1, active(X2)) | | U11(mark(X1), X2, X3) | → | mark(U11(X1, X2, X3)) |
| U12(mark(X1), X2, X3) | → | mark(U12(X1, X2, X3)) | | s(mark(X)) | → | mark(s(X)) |
| plus(mark(X1), X2) | → | mark(plus(X1, X2)) | | plus(X1, mark(X2)) | → | mark(plus(X1, X2)) |
| U21(mark(X1), X2, X3) | → | mark(U21(X1, X2, X3)) | | U22(mark(X1), X2, X3) | → | mark(U22(X1, X2, X3)) |
| x(mark(X1), X2) | → | mark(x(X1, X2)) | | x(X1, mark(X2)) | → | mark(x(X1, X2)) |
| proper(U11(X1, X2, X3)) | → | U11(proper(X1), proper(X2), proper(X3)) | | proper(tt) | → | ok(tt) |
| proper(U12(X1, X2, X3)) | → | U12(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(U21(X1, X2, X3)) | → | U21(proper(X1), proper(X2), proper(X3)) |
| proper(U22(X1, X2, X3)) | → | U22(proper(X1), proper(X2), proper(X3)) | | proper(x(X1, X2)) | → | x(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | U11(ok(X1), ok(X2), ok(X3)) | → | ok(U11(X1, X2, X3)) |
| U12(ok(X1), ok(X2), ok(X3)) | → | ok(U12(X1, X2, X3)) | | s(ok(X)) | → | ok(s(X)) |
| plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) | | U21(ok(X1), ok(X2), ok(X3)) | → | ok(U21(X1, X2, X3)) |
| U22(ok(X1), ok(X2), ok(X3)) | → | ok(U22(X1, X2, X3)) | | x(ok(X1), ok(X2)) | → | ok(x(X1, X2)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, 0, s, tt, active, U11, U12, ok, proper, U21, U22, x, top
Strategy
Function Precedence
plus = mark = 0 = s = tt = U11 = active = U12 = ok = proper = x# = U21 = x = top = U22
Argument Filtering
plus: collapses to 1
mark: 1
0: all arguments are removed from 0
s: all arguments are removed from s
tt: all arguments are removed from tt
U11: 2 3
active: collapses to 1
U12: all arguments are removed from U12
ok: collapses to 1
proper: collapses to 1
x#: collapses to 2
U21: 2 3
x: 1 2
top: all arguments are removed from top
U22: all arguments are removed from U22
Status
mark: multiset
0: multiset
s: multiset
tt: multiset
U11: lexicographic with permutation 2 → 2 3 → 1
U12: multiset
U21: lexicographic with permutation 2 → 2 3 → 1
x: lexicographic with permutation 1 → 2 2 → 1
top: 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.
| x#(X1, mark(X2)) → x#(X1, X2) |
Problem 5: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U21#(ok(X1), ok(X2), ok(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)) |
| active(U11(X1, X2, X3)) | → | U11(active(X1), X2, X3) | | active(U12(X1, X2, X3)) | → | U12(active(X1), X2, X3) |
| active(s(X)) | → | s(active(X)) | | active(plus(X1, X2)) | → | plus(active(X1), X2) |
| active(plus(X1, X2)) | → | plus(X1, active(X2)) | | active(U21(X1, X2, X3)) | → | U21(active(X1), X2, X3) |
| active(U22(X1, X2, X3)) | → | U22(active(X1), X2, X3) | | active(x(X1, X2)) | → | x(active(X1), X2) |
| active(x(X1, X2)) | → | x(X1, active(X2)) | | U11(mark(X1), X2, X3) | → | mark(U11(X1, X2, X3)) |
| U12(mark(X1), X2, X3) | → | mark(U12(X1, X2, X3)) | | s(mark(X)) | → | mark(s(X)) |
| plus(mark(X1), X2) | → | mark(plus(X1, X2)) | | plus(X1, mark(X2)) | → | mark(plus(X1, X2)) |
| U21(mark(X1), X2, X3) | → | mark(U21(X1, X2, X3)) | | U22(mark(X1), X2, X3) | → | mark(U22(X1, X2, X3)) |
| x(mark(X1), X2) | → | mark(x(X1, X2)) | | x(X1, mark(X2)) | → | mark(x(X1, X2)) |
| proper(U11(X1, X2, X3)) | → | U11(proper(X1), proper(X2), proper(X3)) | | proper(tt) | → | ok(tt) |
| proper(U12(X1, X2, X3)) | → | U12(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(U21(X1, X2, X3)) | → | U21(proper(X1), proper(X2), proper(X3)) |
| proper(U22(X1, X2, X3)) | → | U22(proper(X1), proper(X2), proper(X3)) | | proper(x(X1, X2)) | → | x(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | U11(ok(X1), ok(X2), ok(X3)) | → | ok(U11(X1, X2, X3)) |
| U12(ok(X1), ok(X2), ok(X3)) | → | ok(U12(X1, X2, X3)) | | s(ok(X)) | → | ok(s(X)) |
| plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) | | U21(ok(X1), ok(X2), ok(X3)) | → | ok(U21(X1, X2, X3)) |
| U22(ok(X1), ok(X2), ok(X3)) | → | ok(U22(X1, X2, X3)) | | x(ok(X1), ok(X2)) | → | ok(x(X1, X2)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, 0, s, tt, active, U11, U12, ok, proper, U21, x, top, U22
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U21#(ok(X1), ok(X2), ok(X3)) | → | U21#(X1, X2, X3) | | U21#(mark(X1), X2, X3) | → | U21#(X1, X2, X3) |
Problem 6: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| plus#(ok(X1), ok(X2)) | → | plus#(X1, X2) | | plus#(X1, mark(X2)) | → | plus#(X1, X2) |
| plus#(mark(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)) |
| active(U11(X1, X2, X3)) | → | U11(active(X1), X2, X3) | | active(U12(X1, X2, X3)) | → | U12(active(X1), X2, X3) |
| active(s(X)) | → | s(active(X)) | | active(plus(X1, X2)) | → | plus(active(X1), X2) |
| active(plus(X1, X2)) | → | plus(X1, active(X2)) | | active(U21(X1, X2, X3)) | → | U21(active(X1), X2, X3) |
| active(U22(X1, X2, X3)) | → | U22(active(X1), X2, X3) | | active(x(X1, X2)) | → | x(active(X1), X2) |
| active(x(X1, X2)) | → | x(X1, active(X2)) | | U11(mark(X1), X2, X3) | → | mark(U11(X1, X2, X3)) |
| U12(mark(X1), X2, X3) | → | mark(U12(X1, X2, X3)) | | s(mark(X)) | → | mark(s(X)) |
| plus(mark(X1), X2) | → | mark(plus(X1, X2)) | | plus(X1, mark(X2)) | → | mark(plus(X1, X2)) |
| U21(mark(X1), X2, X3) | → | mark(U21(X1, X2, X3)) | | U22(mark(X1), X2, X3) | → | mark(U22(X1, X2, X3)) |
| x(mark(X1), X2) | → | mark(x(X1, X2)) | | x(X1, mark(X2)) | → | mark(x(X1, X2)) |
| proper(U11(X1, X2, X3)) | → | U11(proper(X1), proper(X2), proper(X3)) | | proper(tt) | → | ok(tt) |
| proper(U12(X1, X2, X3)) | → | U12(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(U21(X1, X2, X3)) | → | U21(proper(X1), proper(X2), proper(X3)) |
| proper(U22(X1, X2, X3)) | → | U22(proper(X1), proper(X2), proper(X3)) | | proper(x(X1, X2)) | → | x(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | U11(ok(X1), ok(X2), ok(X3)) | → | ok(U11(X1, X2, X3)) |
| U12(ok(X1), ok(X2), ok(X3)) | → | ok(U12(X1, X2, X3)) | | s(ok(X)) | → | ok(s(X)) |
| plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) | | U21(ok(X1), ok(X2), ok(X3)) | → | ok(U21(X1, X2, X3)) |
| U22(ok(X1), ok(X2), ok(X3)) | → | ok(U22(X1, X2, X3)) | | x(ok(X1), ok(X2)) | → | ok(x(X1, X2)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, 0, s, tt, active, U11, U12, ok, proper, U21, x, top, U22
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| plus#(ok(X1), ok(X2)) | → | plus#(X1, X2) | | plus#(mark(X1), X2) | → | plus#(X1, X2) |
Problem 13: PolynomialLinearRange4iUR
Dependency Pair Problem
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)) |
| active(U11(X1, X2, X3)) | → | U11(active(X1), X2, X3) | | active(U12(X1, X2, X3)) | → | U12(active(X1), X2, X3) |
| active(s(X)) | → | s(active(X)) | | active(plus(X1, X2)) | → | plus(active(X1), X2) |
| active(plus(X1, X2)) | → | plus(X1, active(X2)) | | active(U21(X1, X2, X3)) | → | U21(active(X1), X2, X3) |
| active(U22(X1, X2, X3)) | → | U22(active(X1), X2, X3) | | active(x(X1, X2)) | → | x(active(X1), X2) |
| active(x(X1, X2)) | → | x(X1, active(X2)) | | U11(mark(X1), X2, X3) | → | mark(U11(X1, X2, X3)) |
| U12(mark(X1), X2, X3) | → | mark(U12(X1, X2, X3)) | | s(mark(X)) | → | mark(s(X)) |
| plus(mark(X1), X2) | → | mark(plus(X1, X2)) | | plus(X1, mark(X2)) | → | mark(plus(X1, X2)) |
| U21(mark(X1), X2, X3) | → | mark(U21(X1, X2, X3)) | | U22(mark(X1), X2, X3) | → | mark(U22(X1, X2, X3)) |
| x(mark(X1), X2) | → | mark(x(X1, X2)) | | x(X1, mark(X2)) | → | mark(x(X1, X2)) |
| proper(U11(X1, X2, X3)) | → | U11(proper(X1), proper(X2), proper(X3)) | | proper(tt) | → | ok(tt) |
| proper(U12(X1, X2, X3)) | → | U12(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(U21(X1, X2, X3)) | → | U21(proper(X1), proper(X2), proper(X3)) |
| proper(U22(X1, X2, X3)) | → | U22(proper(X1), proper(X2), proper(X3)) | | proper(x(X1, X2)) | → | x(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | U11(ok(X1), ok(X2), ok(X3)) | → | ok(U11(X1, X2, X3)) |
| U12(ok(X1), ok(X2), ok(X3)) | → | ok(U12(X1, X2, X3)) | | s(ok(X)) | → | ok(s(X)) |
| plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) | | U21(ok(X1), ok(X2), ok(X3)) | → | ok(U21(X1, X2, X3)) |
| U22(ok(X1), ok(X2), ok(X3)) | → | ok(U22(X1, X2, X3)) | | x(ok(X1), ok(X2)) | → | ok(x(X1, X2)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, 0, s, tt, active, U11, U12, ok, proper, U21, U22, x, top
Strategy
Polynomial Interpretation
- 0: 0
- U11(x,y,z): 0
- U12(x,y,z): 0
- U21(x,y,z): 0
- U22(x,y,z): 0
- active(x): 0
- mark(x): 2x + 1
- ok(x): 0
- plus(x,y): 0
- plus#(x,y): 2y + x
- proper(x): 0
- s(x): 0
- top(x): 0
- tt: 0
- x(x,y): 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:
| plus#(X1, mark(X2)) | → | plus#(X1, X2) |
Problem 7: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U22#(mark(X1), X2, X3) | → | U22#(X1, X2, X3) | | U22#(ok(X1), ok(X2), ok(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)) |
| active(U11(X1, X2, X3)) | → | U11(active(X1), X2, X3) | | active(U12(X1, X2, X3)) | → | U12(active(X1), X2, X3) |
| active(s(X)) | → | s(active(X)) | | active(plus(X1, X2)) | → | plus(active(X1), X2) |
| active(plus(X1, X2)) | → | plus(X1, active(X2)) | | active(U21(X1, X2, X3)) | → | U21(active(X1), X2, X3) |
| active(U22(X1, X2, X3)) | → | U22(active(X1), X2, X3) | | active(x(X1, X2)) | → | x(active(X1), X2) |
| active(x(X1, X2)) | → | x(X1, active(X2)) | | U11(mark(X1), X2, X3) | → | mark(U11(X1, X2, X3)) |
| U12(mark(X1), X2, X3) | → | mark(U12(X1, X2, X3)) | | s(mark(X)) | → | mark(s(X)) |
| plus(mark(X1), X2) | → | mark(plus(X1, X2)) | | plus(X1, mark(X2)) | → | mark(plus(X1, X2)) |
| U21(mark(X1), X2, X3) | → | mark(U21(X1, X2, X3)) | | U22(mark(X1), X2, X3) | → | mark(U22(X1, X2, X3)) |
| x(mark(X1), X2) | → | mark(x(X1, X2)) | | x(X1, mark(X2)) | → | mark(x(X1, X2)) |
| proper(U11(X1, X2, X3)) | → | U11(proper(X1), proper(X2), proper(X3)) | | proper(tt) | → | ok(tt) |
| proper(U12(X1, X2, X3)) | → | U12(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(U21(X1, X2, X3)) | → | U21(proper(X1), proper(X2), proper(X3)) |
| proper(U22(X1, X2, X3)) | → | U22(proper(X1), proper(X2), proper(X3)) | | proper(x(X1, X2)) | → | x(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | U11(ok(X1), ok(X2), ok(X3)) | → | ok(U11(X1, X2, X3)) |
| U12(ok(X1), ok(X2), ok(X3)) | → | ok(U12(X1, X2, X3)) | | s(ok(X)) | → | ok(s(X)) |
| plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) | | U21(ok(X1), ok(X2), ok(X3)) | → | ok(U21(X1, X2, X3)) |
| U22(ok(X1), ok(X2), ok(X3)) | → | ok(U22(X1, X2, X3)) | | x(ok(X1), ok(X2)) | → | ok(x(X1, X2)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, 0, s, tt, active, U11, U12, ok, proper, U21, x, top, U22
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U22#(ok(X1), ok(X2), ok(X3)) | → | U22#(X1, X2, X3) | | U22#(mark(X1), X2, X3) | → | U22#(X1, X2, X3) |
Problem 8: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| active#(U11(X1, X2, X3)) | → | active#(X1) | | active#(U22(X1, X2, X3)) | → | active#(X1) |
| active#(U12(X1, X2, X3)) | → | active#(X1) | | active#(plus(X1, X2)) | → | active#(X1) |
| active#(s(X)) | → | active#(X) | | active#(x(X1, X2)) | → | active#(X1) |
| active#(x(X1, X2)) | → | active#(X2) | | active#(plus(X1, X2)) | → | active#(X2) |
| active#(U21(X1, X2, X3)) | → | active#(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)) |
| active(U11(X1, X2, X3)) | → | U11(active(X1), X2, X3) | | active(U12(X1, X2, X3)) | → | U12(active(X1), X2, X3) |
| active(s(X)) | → | s(active(X)) | | active(plus(X1, X2)) | → | plus(active(X1), X2) |
| active(plus(X1, X2)) | → | plus(X1, active(X2)) | | active(U21(X1, X2, X3)) | → | U21(active(X1), X2, X3) |
| active(U22(X1, X2, X3)) | → | U22(active(X1), X2, X3) | | active(x(X1, X2)) | → | x(active(X1), X2) |
| active(x(X1, X2)) | → | x(X1, active(X2)) | | U11(mark(X1), X2, X3) | → | mark(U11(X1, X2, X3)) |
| U12(mark(X1), X2, X3) | → | mark(U12(X1, X2, X3)) | | s(mark(X)) | → | mark(s(X)) |
| plus(mark(X1), X2) | → | mark(plus(X1, X2)) | | plus(X1, mark(X2)) | → | mark(plus(X1, X2)) |
| U21(mark(X1), X2, X3) | → | mark(U21(X1, X2, X3)) | | U22(mark(X1), X2, X3) | → | mark(U22(X1, X2, X3)) |
| x(mark(X1), X2) | → | mark(x(X1, X2)) | | x(X1, mark(X2)) | → | mark(x(X1, X2)) |
| proper(U11(X1, X2, X3)) | → | U11(proper(X1), proper(X2), proper(X3)) | | proper(tt) | → | ok(tt) |
| proper(U12(X1, X2, X3)) | → | U12(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(U21(X1, X2, X3)) | → | U21(proper(X1), proper(X2), proper(X3)) |
| proper(U22(X1, X2, X3)) | → | U22(proper(X1), proper(X2), proper(X3)) | | proper(x(X1, X2)) | → | x(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | U11(ok(X1), ok(X2), ok(X3)) | → | ok(U11(X1, X2, X3)) |
| U12(ok(X1), ok(X2), ok(X3)) | → | ok(U12(X1, X2, X3)) | | s(ok(X)) | → | ok(s(X)) |
| plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) | | U21(ok(X1), ok(X2), ok(X3)) | → | ok(U21(X1, X2, X3)) |
| U22(ok(X1), ok(X2), ok(X3)) | → | ok(U22(X1, X2, X3)) | | x(ok(X1), ok(X2)) | → | ok(x(X1, X2)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, 0, s, tt, active, U11, U12, ok, proper, U21, x, top, U22
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| active#(U11(X1, X2, X3)) | → | active#(X1) | | active#(U22(X1, X2, X3)) | → | active#(X1) |
| active#(U12(X1, X2, X3)) | → | active#(X1) | | active#(plus(X1, X2)) | → | active#(X1) |
| active#(s(X)) | → | active#(X) | | active#(x(X1, X2)) | → | active#(X1) |
| active#(x(X1, X2)) | → | active#(X2) | | active#(plus(X1, X2)) | → | active#(X2) |
| active#(U21(X1, X2, X3)) | → | active#(X1) |
Problem 9: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| proper#(U11(X1, X2, X3)) | → | proper#(X3) | | proper#(U12(X1, X2, X3)) | → | proper#(X2) |
| proper#(U22(X1, X2, X3)) | → | proper#(X2) | | proper#(U21(X1, X2, X3)) | → | proper#(X3) |
| proper#(U22(X1, X2, X3)) | → | proper#(X3) | | proper#(x(X1, X2)) | → | proper#(X1) |
| proper#(U12(X1, X2, X3)) | → | proper#(X3) | | proper#(x(X1, X2)) | → | proper#(X2) |
| proper#(U11(X1, X2, X3)) | → | proper#(X1) | | proper#(s(X)) | → | proper#(X) |
| proper#(U11(X1, X2, X3)) | → | proper#(X2) | | proper#(U21(X1, X2, X3)) | → | proper#(X1) |
| proper#(U22(X1, X2, X3)) | → | proper#(X1) | | proper#(plus(X1, X2)) | → | proper#(X1) |
| proper#(U21(X1, X2, X3)) | → | proper#(X2) | | proper#(plus(X1, X2)) | → | proper#(X2) |
| proper#(U12(X1, X2, X3)) | → | proper#(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)) |
| active(U11(X1, X2, X3)) | → | U11(active(X1), X2, X3) | | active(U12(X1, X2, X3)) | → | U12(active(X1), X2, X3) |
| active(s(X)) | → | s(active(X)) | | active(plus(X1, X2)) | → | plus(active(X1), X2) |
| active(plus(X1, X2)) | → | plus(X1, active(X2)) | | active(U21(X1, X2, X3)) | → | U21(active(X1), X2, X3) |
| active(U22(X1, X2, X3)) | → | U22(active(X1), X2, X3) | | active(x(X1, X2)) | → | x(active(X1), X2) |
| active(x(X1, X2)) | → | x(X1, active(X2)) | | U11(mark(X1), X2, X3) | → | mark(U11(X1, X2, X3)) |
| U12(mark(X1), X2, X3) | → | mark(U12(X1, X2, X3)) | | s(mark(X)) | → | mark(s(X)) |
| plus(mark(X1), X2) | → | mark(plus(X1, X2)) | | plus(X1, mark(X2)) | → | mark(plus(X1, X2)) |
| U21(mark(X1), X2, X3) | → | mark(U21(X1, X2, X3)) | | U22(mark(X1), X2, X3) | → | mark(U22(X1, X2, X3)) |
| x(mark(X1), X2) | → | mark(x(X1, X2)) | | x(X1, mark(X2)) | → | mark(x(X1, X2)) |
| proper(U11(X1, X2, X3)) | → | U11(proper(X1), proper(X2), proper(X3)) | | proper(tt) | → | ok(tt) |
| proper(U12(X1, X2, X3)) | → | U12(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(U21(X1, X2, X3)) | → | U21(proper(X1), proper(X2), proper(X3)) |
| proper(U22(X1, X2, X3)) | → | U22(proper(X1), proper(X2), proper(X3)) | | proper(x(X1, X2)) | → | x(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | U11(ok(X1), ok(X2), ok(X3)) | → | ok(U11(X1, X2, X3)) |
| U12(ok(X1), ok(X2), ok(X3)) | → | ok(U12(X1, X2, X3)) | | s(ok(X)) | → | ok(s(X)) |
| plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) | | U21(ok(X1), ok(X2), ok(X3)) | → | ok(U21(X1, X2, X3)) |
| U22(ok(X1), ok(X2), ok(X3)) | → | ok(U22(X1, X2, X3)) | | x(ok(X1), ok(X2)) | → | ok(x(X1, X2)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, 0, s, tt, active, U11, U12, ok, proper, U21, x, top, U22
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| proper#(U11(X1, X2, X3)) | → | proper#(X3) | | proper#(U12(X1, X2, X3)) | → | proper#(X2) |
| proper#(U22(X1, X2, X3)) | → | proper#(X2) | | proper#(U21(X1, X2, X3)) | → | proper#(X3) |
| proper#(U22(X1, X2, X3)) | → | proper#(X3) | | proper#(x(X1, X2)) | → | proper#(X1) |
| proper#(x(X1, X2)) | → | proper#(X2) | | proper#(U12(X1, X2, X3)) | → | proper#(X3) |
| proper#(U11(X1, X2, X3)) | → | proper#(X1) | | proper#(s(X)) | → | proper#(X) |
| proper#(U11(X1, X2, X3)) | → | proper#(X2) | | proper#(U21(X1, X2, X3)) | → | proper#(X1) |
| proper#(U22(X1, X2, X3)) | → | proper#(X1) | | proper#(plus(X1, X2)) | → | proper#(X1) |
| proper#(U21(X1, X2, X3)) | → | proper#(X2) | | proper#(plus(X1, X2)) | → | proper#(X2) |
| proper#(U12(X1, X2, X3)) | → | proper#(X1) |
Problem 10: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U11#(mark(X1), X2, X3) | → | U11#(X1, X2, X3) | | U11#(ok(X1), ok(X2), ok(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)) |
| active(U11(X1, X2, X3)) | → | U11(active(X1), X2, X3) | | active(U12(X1, X2, X3)) | → | U12(active(X1), X2, X3) |
| active(s(X)) | → | s(active(X)) | | active(plus(X1, X2)) | → | plus(active(X1), X2) |
| active(plus(X1, X2)) | → | plus(X1, active(X2)) | | active(U21(X1, X2, X3)) | → | U21(active(X1), X2, X3) |
| active(U22(X1, X2, X3)) | → | U22(active(X1), X2, X3) | | active(x(X1, X2)) | → | x(active(X1), X2) |
| active(x(X1, X2)) | → | x(X1, active(X2)) | | U11(mark(X1), X2, X3) | → | mark(U11(X1, X2, X3)) |
| U12(mark(X1), X2, X3) | → | mark(U12(X1, X2, X3)) | | s(mark(X)) | → | mark(s(X)) |
| plus(mark(X1), X2) | → | mark(plus(X1, X2)) | | plus(X1, mark(X2)) | → | mark(plus(X1, X2)) |
| U21(mark(X1), X2, X3) | → | mark(U21(X1, X2, X3)) | | U22(mark(X1), X2, X3) | → | mark(U22(X1, X2, X3)) |
| x(mark(X1), X2) | → | mark(x(X1, X2)) | | x(X1, mark(X2)) | → | mark(x(X1, X2)) |
| proper(U11(X1, X2, X3)) | → | U11(proper(X1), proper(X2), proper(X3)) | | proper(tt) | → | ok(tt) |
| proper(U12(X1, X2, X3)) | → | U12(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(U21(X1, X2, X3)) | → | U21(proper(X1), proper(X2), proper(X3)) |
| proper(U22(X1, X2, X3)) | → | U22(proper(X1), proper(X2), proper(X3)) | | proper(x(X1, X2)) | → | x(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | U11(ok(X1), ok(X2), ok(X3)) | → | ok(U11(X1, X2, X3)) |
| U12(ok(X1), ok(X2), ok(X3)) | → | ok(U12(X1, X2, X3)) | | s(ok(X)) | → | ok(s(X)) |
| plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) | | U21(ok(X1), ok(X2), ok(X3)) | → | ok(U21(X1, X2, X3)) |
| U22(ok(X1), ok(X2), ok(X3)) | → | ok(U22(X1, X2, X3)) | | x(ok(X1), ok(X2)) | → | ok(x(X1, X2)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, 0, s, tt, active, U11, U12, ok, proper, U21, x, top, U22
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U11#(mark(X1), X2, X3) | → | U11#(X1, X2, X3) | | U11#(ok(X1), ok(X2), ok(X3)) | → | U11#(X1, X2, X3) |
Problem 11: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U12#(ok(X1), ok(X2), ok(X3)) | → | U12#(X1, X2, X3) | | U12#(mark(X1), 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)) |
| active(U11(X1, X2, X3)) | → | U11(active(X1), X2, X3) | | active(U12(X1, X2, X3)) | → | U12(active(X1), X2, X3) |
| active(s(X)) | → | s(active(X)) | | active(plus(X1, X2)) | → | plus(active(X1), X2) |
| active(plus(X1, X2)) | → | plus(X1, active(X2)) | | active(U21(X1, X2, X3)) | → | U21(active(X1), X2, X3) |
| active(U22(X1, X2, X3)) | → | U22(active(X1), X2, X3) | | active(x(X1, X2)) | → | x(active(X1), X2) |
| active(x(X1, X2)) | → | x(X1, active(X2)) | | U11(mark(X1), X2, X3) | → | mark(U11(X1, X2, X3)) |
| U12(mark(X1), X2, X3) | → | mark(U12(X1, X2, X3)) | | s(mark(X)) | → | mark(s(X)) |
| plus(mark(X1), X2) | → | mark(plus(X1, X2)) | | plus(X1, mark(X2)) | → | mark(plus(X1, X2)) |
| U21(mark(X1), X2, X3) | → | mark(U21(X1, X2, X3)) | | U22(mark(X1), X2, X3) | → | mark(U22(X1, X2, X3)) |
| x(mark(X1), X2) | → | mark(x(X1, X2)) | | x(X1, mark(X2)) | → | mark(x(X1, X2)) |
| proper(U11(X1, X2, X3)) | → | U11(proper(X1), proper(X2), proper(X3)) | | proper(tt) | → | ok(tt) |
| proper(U12(X1, X2, X3)) | → | U12(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(U21(X1, X2, X3)) | → | U21(proper(X1), proper(X2), proper(X3)) |
| proper(U22(X1, X2, X3)) | → | U22(proper(X1), proper(X2), proper(X3)) | | proper(x(X1, X2)) | → | x(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | U11(ok(X1), ok(X2), ok(X3)) | → | ok(U11(X1, X2, X3)) |
| U12(ok(X1), ok(X2), ok(X3)) | → | ok(U12(X1, X2, X3)) | | s(ok(X)) | → | ok(s(X)) |
| plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) | | U21(ok(X1), ok(X2), ok(X3)) | → | ok(U21(X1, X2, X3)) |
| U22(ok(X1), ok(X2), ok(X3)) | → | ok(U22(X1, X2, X3)) | | x(ok(X1), ok(X2)) | → | ok(x(X1, X2)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: plus, mark, 0, s, tt, active, U11, U12, ok, proper, U21, x, top, U22
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U12#(ok(X1), ok(X2), ok(X3)) | → | U12#(X1, X2, X3) | | U12#(mark(X1), X2, X3) | → | U12#(X1, X2, X3) |