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 (1091ms).
| – Problem 2 was processed with processor SubtermCriterion (1ms).
| – Problem 3 was processed with processor SubtermCriterion (2ms).
| – Problem 4 remains open; application of the following processors failed [SubtermCriterion (2ms), DependencyGraph (190ms), PolynomialLinearRange4iUR (5000ms), DependencyGraph (151ms), PolynomialLinearRange8NegiUR (15000ms), DependencyGraph (142ms), ReductionPairSAT (5574ms), DependencyGraph (143ms), ReductionPairSAT (5612ms), DependencyGraph (172ms), ReductionPairSAT (5490ms), DependencyGraph (151ms), SizeChangePrinciple (timeout)].
| – Problem 5 was processed with processor SubtermCriterion (1ms).
| – Problem 6 was processed with processor SubtermCriterion (1ms).
| – Problem 7 was processed with processor SubtermCriterion (0ms).
| | – Problem 9 was processed with processor ReductionPairSAT (46ms).
| | | – Problem 10 was processed with processor ReductionPairSAT (31ms).
| – Problem 8 was processed with processor SubtermCriterion (1ms).
The following open problems remain:
Open Dependency Pair Problem 4
Dependency Pairs
| mark#(hd(X)) | → | active#(hd(mark(X))) | | mark#(0) | → | active#(0) |
| active#(adx(cons(X, Y))) | → | mark#(incr(cons(X, adx(Y)))) | | mark#(nats) | → | active#(nats) |
| mark#(hd(X)) | → | mark#(X) | | active#(incr(cons(X, Y))) | → | mark#(cons(s(X), incr(Y))) |
| active#(tl(cons(X, Y))) | → | mark#(Y) | | mark#(incr(X)) | → | active#(incr(mark(X))) |
| mark#(incr(X)) | → | mark#(X) | | mark#(s(X)) | → | active#(s(X)) |
| mark#(cons(X1, X2)) | → | active#(cons(X1, X2)) | | mark#(tl(X)) | → | active#(tl(mark(X))) |
| mark#(tl(X)) | → | mark#(X) | | active#(hd(cons(X, Y))) | → | mark#(X) |
| mark#(adx(X)) | → | mark#(X) | | active#(zeros) | → | mark#(cons(0, zeros)) |
| active#(nats) | → | mark#(adx(zeros)) | | mark#(zeros) | → | active#(zeros) |
| mark#(adx(X)) | → | active#(adx(mark(X))) |
Rewrite Rules
| active(nats) | → | mark(adx(zeros)) | | active(zeros) | → | mark(cons(0, zeros)) |
| active(incr(cons(X, Y))) | → | mark(cons(s(X), incr(Y))) | | active(adx(cons(X, Y))) | → | mark(incr(cons(X, adx(Y)))) |
| active(hd(cons(X, Y))) | → | mark(X) | | active(tl(cons(X, Y))) | → | mark(Y) |
| mark(nats) | → | active(nats) | | mark(adx(X)) | → | active(adx(mark(X))) |
| mark(zeros) | → | active(zeros) | | mark(cons(X1, X2)) | → | active(cons(X1, X2)) |
| mark(0) | → | active(0) | | mark(incr(X)) | → | active(incr(mark(X))) |
| mark(s(X)) | → | active(s(X)) | | mark(hd(X)) | → | active(hd(mark(X))) |
| mark(tl(X)) | → | active(tl(mark(X))) | | adx(mark(X)) | → | adx(X) |
| adx(active(X)) | → | adx(X) | | cons(mark(X1), X2) | → | cons(X1, X2) |
| cons(X1, mark(X2)) | → | cons(X1, X2) | | cons(active(X1), X2) | → | cons(X1, X2) |
| cons(X1, active(X2)) | → | cons(X1, X2) | | incr(mark(X)) | → | incr(X) |
| incr(active(X)) | → | incr(X) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | hd(mark(X)) | → | hd(X) |
| hd(active(X)) | → | hd(X) | | tl(mark(X)) | → | tl(X) |
| tl(active(X)) | → | tl(X) |
Original Signature
Termination of terms over the following signature is verified: nats, tl, 0, s, zeros, active, adx, hd, mark, incr, cons
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| hd#(active(X)) | → | hd#(X) | | mark#(hd(X)) | → | active#(hd(mark(X))) |
| mark#(tl(X)) | → | tl#(mark(X)) | | incr#(active(X)) | → | incr#(X) |
| active#(adx(cons(X, Y))) | → | cons#(X, adx(Y)) | | active#(incr(cons(X, Y))) | → | cons#(s(X), incr(Y)) |
| mark#(hd(X)) | → | mark#(X) | | tl#(active(X)) | → | tl#(X) |
| active#(incr(cons(X, Y))) | → | mark#(cons(s(X), incr(Y))) | | cons#(mark(X1), X2) | → | cons#(X1, X2) |
| mark#(cons(X1, X2)) | → | cons#(X1, X2) | | mark#(tl(X)) | → | active#(tl(mark(X))) |
| active#(hd(cons(X, Y))) | → | mark#(X) | | mark#(hd(X)) | → | hd#(mark(X)) |
| incr#(mark(X)) | → | incr#(X) | | active#(nats) | → | mark#(adx(zeros)) |
| cons#(X1, mark(X2)) | → | cons#(X1, X2) | | mark#(incr(X)) | → | incr#(mark(X)) |
| mark#(adx(X)) | → | adx#(mark(X)) | | mark#(zeros) | → | active#(zeros) |
| active#(adx(cons(X, Y))) | → | adx#(Y) | | active#(adx(cons(X, Y))) | → | mark#(incr(cons(X, adx(Y)))) |
| active#(incr(cons(X, Y))) | → | incr#(Y) | | mark#(0) | → | active#(0) |
| adx#(active(X)) | → | adx#(X) | | active#(adx(cons(X, Y))) | → | incr#(cons(X, adx(Y))) |
| mark#(nats) | → | active#(nats) | | hd#(mark(X)) | → | hd#(X) |
| active#(tl(cons(X, Y))) | → | mark#(Y) | | cons#(active(X1), X2) | → | cons#(X1, X2) |
| mark#(incr(X)) | → | active#(incr(mark(X))) | | mark#(incr(X)) | → | mark#(X) |
| mark#(s(X)) | → | active#(s(X)) | | mark#(cons(X1, X2)) | → | active#(cons(X1, X2)) |
| active#(nats) | → | adx#(zeros) | | mark#(s(X)) | → | s#(X) |
| s#(mark(X)) | → | s#(X) | | adx#(mark(X)) | → | adx#(X) |
| cons#(X1, active(X2)) | → | cons#(X1, X2) | | active#(incr(cons(X, Y))) | → | s#(X) |
| mark#(tl(X)) | → | mark#(X) | | mark#(adx(X)) | → | mark#(X) |
| tl#(mark(X)) | → | tl#(X) | | active#(zeros) | → | cons#(0, zeros) |
| active#(zeros) | → | mark#(cons(0, zeros)) | | s#(active(X)) | → | s#(X) |
| mark#(adx(X)) | → | active#(adx(mark(X))) |
Rewrite Rules
| active(nats) | → | mark(adx(zeros)) | | active(zeros) | → | mark(cons(0, zeros)) |
| active(incr(cons(X, Y))) | → | mark(cons(s(X), incr(Y))) | | active(adx(cons(X, Y))) | → | mark(incr(cons(X, adx(Y)))) |
| active(hd(cons(X, Y))) | → | mark(X) | | active(tl(cons(X, Y))) | → | mark(Y) |
| mark(nats) | → | active(nats) | | mark(adx(X)) | → | active(adx(mark(X))) |
| mark(zeros) | → | active(zeros) | | mark(cons(X1, X2)) | → | active(cons(X1, X2)) |
| mark(0) | → | active(0) | | mark(incr(X)) | → | active(incr(mark(X))) |
| mark(s(X)) | → | active(s(X)) | | mark(hd(X)) | → | active(hd(mark(X))) |
| mark(tl(X)) | → | active(tl(mark(X))) | | adx(mark(X)) | → | adx(X) |
| adx(active(X)) | → | adx(X) | | cons(mark(X1), X2) | → | cons(X1, X2) |
| cons(X1, mark(X2)) | → | cons(X1, X2) | | cons(active(X1), X2) | → | cons(X1, X2) |
| cons(X1, active(X2)) | → | cons(X1, X2) | | incr(mark(X)) | → | incr(X) |
| incr(active(X)) | → | incr(X) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | hd(mark(X)) | → | hd(X) |
| hd(active(X)) | → | hd(X) | | tl(mark(X)) | → | tl(X) |
| tl(active(X)) | → | tl(X) |
Original Signature
Termination of terms over the following signature is verified: nats, 0, tl, s, zeros, adx, active, mark, hd, incr, cons
Strategy
The following SCCs where found
| adx#(mark(X)) → adx#(X) | adx#(active(X)) → adx#(X) |
| hd#(active(X)) → hd#(X) | hd#(mark(X)) → hd#(X) |
| tl#(active(X)) → tl#(X) | tl#(mark(X)) → tl#(X) |
| cons#(X1, active(X2)) → cons#(X1, X2) | cons#(mark(X1), X2) → cons#(X1, X2) |
| cons#(X1, mark(X2)) → cons#(X1, X2) | cons#(active(X1), X2) → cons#(X1, X2) |
| s#(mark(X)) → s#(X) | s#(active(X)) → s#(X) |
| incr#(active(X)) → incr#(X) | incr#(mark(X)) → incr#(X) |
| active#(adx(cons(X, Y))) → mark#(incr(cons(X, adx(Y)))) | mark#(0) → active#(0) |
| mark#(hd(X)) → active#(hd(mark(X))) | mark#(nats) → active#(nats) |
| mark#(hd(X)) → mark#(X) | active#(incr(cons(X, Y))) → mark#(cons(s(X), incr(Y))) |
| active#(tl(cons(X, Y))) → mark#(Y) | mark#(incr(X)) → active#(incr(mark(X))) |
| mark#(incr(X)) → mark#(X) | mark#(s(X)) → active#(s(X)) |
| mark#(cons(X1, X2)) → active#(cons(X1, X2)) | mark#(tl(X)) → active#(tl(mark(X))) |
| active#(hd(cons(X, Y))) → mark#(X) | mark#(tl(X)) → mark#(X) |
| mark#(adx(X)) → mark#(X) | active#(nats) → mark#(adx(zeros)) |
| active#(zeros) → mark#(cons(0, zeros)) | mark#(zeros) → active#(zeros) |
| mark#(adx(X)) → active#(adx(mark(X))) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| adx#(mark(X)) | → | adx#(X) | | adx#(active(X)) | → | adx#(X) |
Rewrite Rules
| active(nats) | → | mark(adx(zeros)) | | active(zeros) | → | mark(cons(0, zeros)) |
| active(incr(cons(X, Y))) | → | mark(cons(s(X), incr(Y))) | | active(adx(cons(X, Y))) | → | mark(incr(cons(X, adx(Y)))) |
| active(hd(cons(X, Y))) | → | mark(X) | | active(tl(cons(X, Y))) | → | mark(Y) |
| mark(nats) | → | active(nats) | | mark(adx(X)) | → | active(adx(mark(X))) |
| mark(zeros) | → | active(zeros) | | mark(cons(X1, X2)) | → | active(cons(X1, X2)) |
| mark(0) | → | active(0) | | mark(incr(X)) | → | active(incr(mark(X))) |
| mark(s(X)) | → | active(s(X)) | | mark(hd(X)) | → | active(hd(mark(X))) |
| mark(tl(X)) | → | active(tl(mark(X))) | | adx(mark(X)) | → | adx(X) |
| adx(active(X)) | → | adx(X) | | cons(mark(X1), X2) | → | cons(X1, X2) |
| cons(X1, mark(X2)) | → | cons(X1, X2) | | cons(active(X1), X2) | → | cons(X1, X2) |
| cons(X1, active(X2)) | → | cons(X1, X2) | | incr(mark(X)) | → | incr(X) |
| incr(active(X)) | → | incr(X) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | hd(mark(X)) | → | hd(X) |
| hd(active(X)) | → | hd(X) | | tl(mark(X)) | → | tl(X) |
| tl(active(X)) | → | tl(X) |
Original Signature
Termination of terms over the following signature is verified: nats, 0, tl, s, zeros, adx, active, mark, hd, incr, cons
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| adx#(mark(X)) | → | adx#(X) | | adx#(active(X)) | → | adx#(X) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| tl#(active(X)) | → | tl#(X) | | tl#(mark(X)) | → | tl#(X) |
Rewrite Rules
| active(nats) | → | mark(adx(zeros)) | | active(zeros) | → | mark(cons(0, zeros)) |
| active(incr(cons(X, Y))) | → | mark(cons(s(X), incr(Y))) | | active(adx(cons(X, Y))) | → | mark(incr(cons(X, adx(Y)))) |
| active(hd(cons(X, Y))) | → | mark(X) | | active(tl(cons(X, Y))) | → | mark(Y) |
| mark(nats) | → | active(nats) | | mark(adx(X)) | → | active(adx(mark(X))) |
| mark(zeros) | → | active(zeros) | | mark(cons(X1, X2)) | → | active(cons(X1, X2)) |
| mark(0) | → | active(0) | | mark(incr(X)) | → | active(incr(mark(X))) |
| mark(s(X)) | → | active(s(X)) | | mark(hd(X)) | → | active(hd(mark(X))) |
| mark(tl(X)) | → | active(tl(mark(X))) | | adx(mark(X)) | → | adx(X) |
| adx(active(X)) | → | adx(X) | | cons(mark(X1), X2) | → | cons(X1, X2) |
| cons(X1, mark(X2)) | → | cons(X1, X2) | | cons(active(X1), X2) | → | cons(X1, X2) |
| cons(X1, active(X2)) | → | cons(X1, X2) | | incr(mark(X)) | → | incr(X) |
| incr(active(X)) | → | incr(X) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | hd(mark(X)) | → | hd(X) |
| hd(active(X)) | → | hd(X) | | tl(mark(X)) | → | tl(X) |
| tl(active(X)) | → | tl(X) |
Original Signature
Termination of terms over the following signature is verified: nats, 0, tl, s, zeros, adx, active, mark, hd, incr, cons
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| tl#(active(X)) | → | tl#(X) | | tl#(mark(X)) | → | tl#(X) |
Problem 5: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| incr#(active(X)) | → | incr#(X) | | incr#(mark(X)) | → | incr#(X) |
Rewrite Rules
| active(nats) | → | mark(adx(zeros)) | | active(zeros) | → | mark(cons(0, zeros)) |
| active(incr(cons(X, Y))) | → | mark(cons(s(X), incr(Y))) | | active(adx(cons(X, Y))) | → | mark(incr(cons(X, adx(Y)))) |
| active(hd(cons(X, Y))) | → | mark(X) | | active(tl(cons(X, Y))) | → | mark(Y) |
| mark(nats) | → | active(nats) | | mark(adx(X)) | → | active(adx(mark(X))) |
| mark(zeros) | → | active(zeros) | | mark(cons(X1, X2)) | → | active(cons(X1, X2)) |
| mark(0) | → | active(0) | | mark(incr(X)) | → | active(incr(mark(X))) |
| mark(s(X)) | → | active(s(X)) | | mark(hd(X)) | → | active(hd(mark(X))) |
| mark(tl(X)) | → | active(tl(mark(X))) | | adx(mark(X)) | → | adx(X) |
| adx(active(X)) | → | adx(X) | | cons(mark(X1), X2) | → | cons(X1, X2) |
| cons(X1, mark(X2)) | → | cons(X1, X2) | | cons(active(X1), X2) | → | cons(X1, X2) |
| cons(X1, active(X2)) | → | cons(X1, X2) | | incr(mark(X)) | → | incr(X) |
| incr(active(X)) | → | incr(X) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | hd(mark(X)) | → | hd(X) |
| hd(active(X)) | → | hd(X) | | tl(mark(X)) | → | tl(X) |
| tl(active(X)) | → | tl(X) |
Original Signature
Termination of terms over the following signature is verified: nats, 0, tl, s, zeros, adx, active, mark, hd, incr, cons
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| incr#(active(X)) | → | incr#(X) | | incr#(mark(X)) | → | incr#(X) |
Problem 6: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| s#(mark(X)) | → | s#(X) | | s#(active(X)) | → | s#(X) |
Rewrite Rules
| active(nats) | → | mark(adx(zeros)) | | active(zeros) | → | mark(cons(0, zeros)) |
| active(incr(cons(X, Y))) | → | mark(cons(s(X), incr(Y))) | | active(adx(cons(X, Y))) | → | mark(incr(cons(X, adx(Y)))) |
| active(hd(cons(X, Y))) | → | mark(X) | | active(tl(cons(X, Y))) | → | mark(Y) |
| mark(nats) | → | active(nats) | | mark(adx(X)) | → | active(adx(mark(X))) |
| mark(zeros) | → | active(zeros) | | mark(cons(X1, X2)) | → | active(cons(X1, X2)) |
| mark(0) | → | active(0) | | mark(incr(X)) | → | active(incr(mark(X))) |
| mark(s(X)) | → | active(s(X)) | | mark(hd(X)) | → | active(hd(mark(X))) |
| mark(tl(X)) | → | active(tl(mark(X))) | | adx(mark(X)) | → | adx(X) |
| adx(active(X)) | → | adx(X) | | cons(mark(X1), X2) | → | cons(X1, X2) |
| cons(X1, mark(X2)) | → | cons(X1, X2) | | cons(active(X1), X2) | → | cons(X1, X2) |
| cons(X1, active(X2)) | → | cons(X1, X2) | | incr(mark(X)) | → | incr(X) |
| incr(active(X)) | → | incr(X) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | hd(mark(X)) | → | hd(X) |
| hd(active(X)) | → | hd(X) | | tl(mark(X)) | → | tl(X) |
| tl(active(X)) | → | tl(X) |
Original Signature
Termination of terms over the following signature is verified: nats, 0, tl, s, zeros, adx, active, mark, hd, incr, cons
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| s#(mark(X)) | → | s#(X) | | s#(active(X)) | → | s#(X) |
Problem 7: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| cons#(X1, active(X2)) | → | cons#(X1, X2) | | cons#(mark(X1), X2) | → | cons#(X1, X2) |
| cons#(X1, mark(X2)) | → | cons#(X1, X2) | | cons#(active(X1), X2) | → | cons#(X1, X2) |
Rewrite Rules
| active(nats) | → | mark(adx(zeros)) | | active(zeros) | → | mark(cons(0, zeros)) |
| active(incr(cons(X, Y))) | → | mark(cons(s(X), incr(Y))) | | active(adx(cons(X, Y))) | → | mark(incr(cons(X, adx(Y)))) |
| active(hd(cons(X, Y))) | → | mark(X) | | active(tl(cons(X, Y))) | → | mark(Y) |
| mark(nats) | → | active(nats) | | mark(adx(X)) | → | active(adx(mark(X))) |
| mark(zeros) | → | active(zeros) | | mark(cons(X1, X2)) | → | active(cons(X1, X2)) |
| mark(0) | → | active(0) | | mark(incr(X)) | → | active(incr(mark(X))) |
| mark(s(X)) | → | active(s(X)) | | mark(hd(X)) | → | active(hd(mark(X))) |
| mark(tl(X)) | → | active(tl(mark(X))) | | adx(mark(X)) | → | adx(X) |
| adx(active(X)) | → | adx(X) | | cons(mark(X1), X2) | → | cons(X1, X2) |
| cons(X1, mark(X2)) | → | cons(X1, X2) | | cons(active(X1), X2) | → | cons(X1, X2) |
| cons(X1, active(X2)) | → | cons(X1, X2) | | incr(mark(X)) | → | incr(X) |
| incr(active(X)) | → | incr(X) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | hd(mark(X)) | → | hd(X) |
| hd(active(X)) | → | hd(X) | | tl(mark(X)) | → | tl(X) |
| tl(active(X)) | → | tl(X) |
Original Signature
Termination of terms over the following signature is verified: nats, 0, tl, s, zeros, adx, active, mark, hd, incr, cons
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| cons#(mark(X1), X2) | → | cons#(X1, X2) | | cons#(active(X1), X2) | → | cons#(X1, X2) |
Problem 9: ReductionPairSAT
Dependency Pair Problem
Dependency Pairs
| cons#(X1, active(X2)) | → | cons#(X1, X2) | | cons#(X1, mark(X2)) | → | cons#(X1, X2) |
Rewrite Rules
| active(nats) | → | mark(adx(zeros)) | | active(zeros) | → | mark(cons(0, zeros)) |
| active(incr(cons(X, Y))) | → | mark(cons(s(X), incr(Y))) | | active(adx(cons(X, Y))) | → | mark(incr(cons(X, adx(Y)))) |
| active(hd(cons(X, Y))) | → | mark(X) | | active(tl(cons(X, Y))) | → | mark(Y) |
| mark(nats) | → | active(nats) | | mark(adx(X)) | → | active(adx(mark(X))) |
| mark(zeros) | → | active(zeros) | | mark(cons(X1, X2)) | → | active(cons(X1, X2)) |
| mark(0) | → | active(0) | | mark(incr(X)) | → | active(incr(mark(X))) |
| mark(s(X)) | → | active(s(X)) | | mark(hd(X)) | → | active(hd(mark(X))) |
| mark(tl(X)) | → | active(tl(mark(X))) | | adx(mark(X)) | → | adx(X) |
| adx(active(X)) | → | adx(X) | | cons(mark(X1), X2) | → | cons(X1, X2) |
| cons(X1, mark(X2)) | → | cons(X1, X2) | | cons(active(X1), X2) | → | cons(X1, X2) |
| cons(X1, active(X2)) | → | cons(X1, X2) | | incr(mark(X)) | → | incr(X) |
| incr(active(X)) | → | incr(X) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | hd(mark(X)) | → | hd(X) |
| hd(active(X)) | → | hd(X) | | tl(mark(X)) | → | tl(X) |
| tl(active(X)) | → | tl(X) |
Original Signature
Termination of terms over the following signature is verified: nats, tl, 0, s, zeros, active, adx, hd, mark, incr, cons
Strategy
Function Precedence
active = mark < cons# = nats = tl = 0 = s = zeros = adx = hd = incr = cons
Argument Filtering
cons#: collapses to 2
nats: all arguments are removed from nats
tl: 1
0: all arguments are removed from 0
s: all arguments are removed from s
zeros: all arguments are removed from zeros
active: 1
adx: all arguments are removed from adx
mark: collapses to 1
hd: all arguments are removed from hd
incr: all arguments are removed from incr
cons: 1 2
Status
nats: multiset
tl: lexicographic with permutation 1 → 1
0: multiset
s: multiset
zeros: multiset
active: multiset
adx: multiset
hd: multiset
incr: multiset
cons: lexicographic with permutation 1 → 1 2 → 2
Usable Rules
There are no usable rules.
The dependency pairs and usable rules are stronlgy conservative!
Eliminated dependency pairs
The following dependency pairs (at least) can be eliminated according to the given precedence.
| cons#(X1, active(X2)) → cons#(X1, X2) |
Problem 10: ReductionPairSAT
Dependency Pair Problem
Dependency Pairs
| cons#(X1, mark(X2)) | → | cons#(X1, X2) |
Rewrite Rules
| active(nats) | → | mark(adx(zeros)) | | active(zeros) | → | mark(cons(0, zeros)) |
| active(incr(cons(X, Y))) | → | mark(cons(s(X), incr(Y))) | | active(adx(cons(X, Y))) | → | mark(incr(cons(X, adx(Y)))) |
| active(hd(cons(X, Y))) | → | mark(X) | | active(tl(cons(X, Y))) | → | mark(Y) |
| mark(nats) | → | active(nats) | | mark(adx(X)) | → | active(adx(mark(X))) |
| mark(zeros) | → | active(zeros) | | mark(cons(X1, X2)) | → | active(cons(X1, X2)) |
| mark(0) | → | active(0) | | mark(incr(X)) | → | active(incr(mark(X))) |
| mark(s(X)) | → | active(s(X)) | | mark(hd(X)) | → | active(hd(mark(X))) |
| mark(tl(X)) | → | active(tl(mark(X))) | | adx(mark(X)) | → | adx(X) |
| adx(active(X)) | → | adx(X) | | cons(mark(X1), X2) | → | cons(X1, X2) |
| cons(X1, mark(X2)) | → | cons(X1, X2) | | cons(active(X1), X2) | → | cons(X1, X2) |
| cons(X1, active(X2)) | → | cons(X1, X2) | | incr(mark(X)) | → | incr(X) |
| incr(active(X)) | → | incr(X) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | hd(mark(X)) | → | hd(X) |
| hd(active(X)) | → | hd(X) | | tl(mark(X)) | → | tl(X) |
| tl(active(X)) | → | tl(X) |
Original Signature
Termination of terms over the following signature is verified: nats, 0, tl, s, zeros, adx, active, mark, hd, incr, cons
Strategy
Function Precedence
cons# = nats = tl = 0 = s = zeros = active = adx = mark = hd = incr = cons
Argument Filtering
cons#: collapses to 2
nats: all arguments are removed from nats
tl: all arguments are removed from tl
0: all arguments are removed from 0
s: all arguments are removed from s
zeros: all arguments are removed from zeros
active: all arguments are removed from active
adx: 1
mark: 1
hd: 1
incr: all arguments are removed from incr
cons: 2
Status
nats: multiset
tl: multiset
0: multiset
s: multiset
zeros: multiset
active: multiset
adx: lexicographic with permutation 1 → 1
mark: lexicographic with permutation 1 → 1
hd: lexicographic with permutation 1 → 1
incr: multiset
cons: lexicographic with permutation 2 → 1
Usable Rules
There are no usable rules.
The dependency pairs and usable rules are stronlgy conservative!
Eliminated dependency pairs
The following dependency pairs (at least) can be eliminated according to the given precedence.
| cons#(X1, mark(X2)) → cons#(X1, X2) |
Problem 8: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| hd#(active(X)) | → | hd#(X) | | hd#(mark(X)) | → | hd#(X) |
Rewrite Rules
| active(nats) | → | mark(adx(zeros)) | | active(zeros) | → | mark(cons(0, zeros)) |
| active(incr(cons(X, Y))) | → | mark(cons(s(X), incr(Y))) | | active(adx(cons(X, Y))) | → | mark(incr(cons(X, adx(Y)))) |
| active(hd(cons(X, Y))) | → | mark(X) | | active(tl(cons(X, Y))) | → | mark(Y) |
| mark(nats) | → | active(nats) | | mark(adx(X)) | → | active(adx(mark(X))) |
| mark(zeros) | → | active(zeros) | | mark(cons(X1, X2)) | → | active(cons(X1, X2)) |
| mark(0) | → | active(0) | | mark(incr(X)) | → | active(incr(mark(X))) |
| mark(s(X)) | → | active(s(X)) | | mark(hd(X)) | → | active(hd(mark(X))) |
| mark(tl(X)) | → | active(tl(mark(X))) | | adx(mark(X)) | → | adx(X) |
| adx(active(X)) | → | adx(X) | | cons(mark(X1), X2) | → | cons(X1, X2) |
| cons(X1, mark(X2)) | → | cons(X1, X2) | | cons(active(X1), X2) | → | cons(X1, X2) |
| cons(X1, active(X2)) | → | cons(X1, X2) | | incr(mark(X)) | → | incr(X) |
| incr(active(X)) | → | incr(X) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | hd(mark(X)) | → | hd(X) |
| hd(active(X)) | → | hd(X) | | tl(mark(X)) | → | tl(X) |
| tl(active(X)) | → | tl(X) |
Original Signature
Termination of terms over the following signature is verified: nats, 0, tl, s, zeros, adx, active, mark, hd, incr, cons
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| hd#(active(X)) | → | hd#(X) | | hd#(mark(X)) | → | hd#(X) |