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 (5529ms).
| – Problem 2 was processed with processor SubtermCriterion (1ms).
| – Problem 3 was processed with processor SubtermCriterion (4ms).
| – Problem 4 was processed with processor SubtermCriterion (1ms).
| | – Problem 13 was processed with processor ReductionPairSAT (76ms).
| – Problem 5 was processed with processor SubtermCriterion (2ms).
| – Problem 6 was processed with processor SubtermCriterion (1ms).
| – Problem 7 was processed with processor SubtermCriterion (3ms).
| – Problem 8 was processed with processor SubtermCriterion (1ms).
| | – Problem 14 was processed with processor ReductionPairSAT (83ms).
| – Problem 9 remains open; application of the following processors failed [SubtermCriterion (1ms), DependencyGraph (7ms), PolynomialLinearRange4iUR (2500ms), DependencyGraph (6ms), PolynomialLinearRange4iUR (3348ms), DependencyGraph (4ms), PolynomialLinearRange8NegiUR (10000ms), DependencyGraph (4ms), ReductionPairSAT (10325ms), DependencyGraph (5ms), ReductionPairSAT (10420ms), DependencyGraph (44ms), SizeChangePrinciple (timeout)].
| – Problem 10 was processed with processor SubtermCriterion (2ms).
| – Problem 11 was processed with processor SubtermCriterion (1ms).
| | – Problem 15 was processed with processor PolynomialLinearRange4iUR (21ms).
| – Problem 12 was processed with processor SubtermCriterion (3ms).
The following open problems remain:
Open Dependency Pair Problem 9
Dependency Pairs
| top#(mark(X)) | → | top#(proper(X)) | | top#(ok(X)) | → | top#(active(X)) |
Rewrite Rules
| active(pairNs) | → | mark(cons(0, incr(oddNs))) | | active(oddNs) | → | mark(incr(pairNs)) |
| active(incr(cons(X, XS))) | → | mark(cons(s(X), incr(XS))) | | active(take(0, XS)) | → | mark(nil) |
| active(take(s(N), cons(X, XS))) | → | mark(cons(X, take(N, XS))) | | active(zip(nil, XS)) | → | mark(nil) |
| active(zip(X, nil)) | → | mark(nil) | | active(zip(cons(X, XS), cons(Y, YS))) | → | mark(cons(pair(X, Y), zip(XS, YS))) |
| active(tail(cons(X, XS))) | → | mark(XS) | | active(repItems(nil)) | → | mark(nil) |
| active(repItems(cons(X, XS))) | → | mark(cons(X, cons(X, repItems(XS)))) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(incr(X)) | → | incr(active(X)) | | active(s(X)) | → | s(active(X)) |
| active(take(X1, X2)) | → | take(active(X1), X2) | | active(take(X1, X2)) | → | take(X1, active(X2)) |
| active(zip(X1, X2)) | → | zip(active(X1), X2) | | active(zip(X1, X2)) | → | zip(X1, active(X2)) |
| active(pair(X1, X2)) | → | pair(active(X1), X2) | | active(pair(X1, X2)) | → | pair(X1, active(X2)) |
| active(tail(X)) | → | tail(active(X)) | | active(repItems(X)) | → | repItems(active(X)) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | incr(mark(X)) | → | mark(incr(X)) |
| s(mark(X)) | → | mark(s(X)) | | take(mark(X1), X2) | → | mark(take(X1, X2)) |
| take(X1, mark(X2)) | → | mark(take(X1, X2)) | | zip(mark(X1), X2) | → | mark(zip(X1, X2)) |
| zip(X1, mark(X2)) | → | mark(zip(X1, X2)) | | pair(mark(X1), X2) | → | mark(pair(X1, X2)) |
| pair(X1, mark(X2)) | → | mark(pair(X1, X2)) | | tail(mark(X)) | → | mark(tail(X)) |
| repItems(mark(X)) | → | mark(repItems(X)) | | proper(pairNs) | → | ok(pairNs) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(incr(X)) | → | incr(proper(X)) | | proper(oddNs) | → | ok(oddNs) |
| proper(s(X)) | → | s(proper(X)) | | proper(take(X1, X2)) | → | take(proper(X1), proper(X2)) |
| proper(nil) | → | ok(nil) | | proper(zip(X1, X2)) | → | zip(proper(X1), proper(X2)) |
| proper(pair(X1, X2)) | → | pair(proper(X1), proper(X2)) | | proper(tail(X)) | → | tail(proper(X)) |
| proper(repItems(X)) | → | repItems(proper(X)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| incr(ok(X)) | → | ok(incr(X)) | | s(ok(X)) | → | ok(s(X)) |
| take(ok(X1), ok(X2)) | → | ok(take(X1, X2)) | | zip(ok(X1), ok(X2)) | → | ok(zip(X1, X2)) |
| pair(ok(X1), ok(X2)) | → | ok(pair(X1, X2)) | | tail(ok(X)) | → | ok(tail(X)) |
| repItems(ok(X)) | → | ok(repItems(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: zip, pair, mark, tail, 0, pairNs, s, repItems, take, active, ok, proper, incr, oddNs, top, cons, nil
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| active#(zip(cons(X, XS), cons(Y, YS))) | → | pair#(X, Y) | | proper#(cons(X1, X2)) | → | proper#(X1) |
| top#(ok(X)) | → | top#(active(X)) | | incr#(ok(X)) | → | incr#(X) |
| proper#(tail(X)) | → | proper#(X) | | active#(repItems(X)) | → | repItems#(active(X)) |
| cons#(ok(X1), ok(X2)) | → | cons#(X1, X2) | | active#(zip(X1, X2)) | → | zip#(X1, active(X2)) |
| active#(cons(X1, X2)) | → | cons#(active(X1), X2) | | active#(pairNs) | → | incr#(oddNs) |
| active#(repItems(cons(X, XS))) | → | cons#(X, repItems(XS)) | | active#(tail(X)) | → | tail#(active(X)) |
| active#(take(s(N), cons(X, XS))) | → | cons#(X, take(N, XS)) | | tail#(ok(X)) | → | tail#(X) |
| active#(take(X1, X2)) | → | take#(active(X1), X2) | | active#(pair(X1, X2)) | → | active#(X2) |
| proper#(incr(X)) | → | proper#(X) | | active#(pairNs) | → | cons#(0, incr(oddNs)) |
| zip#(ok(X1), ok(X2)) | → | zip#(X1, X2) | | active#(zip(X1, X2)) | → | active#(X1) |
| top#(mark(X)) | → | proper#(X) | | top#(mark(X)) | → | top#(proper(X)) |
| active#(incr(cons(X, XS))) | → | cons#(s(X), incr(XS)) | | proper#(cons(X1, X2)) | → | proper#(X2) |
| proper#(zip(X1, X2)) | → | zip#(proper(X1), proper(X2)) | | active#(take(X1, X2)) | → | active#(X2) |
| proper#(pair(X1, X2)) | → | proper#(X2) | | active#(pair(X1, X2)) | → | pair#(active(X1), X2) |
| pair#(X1, mark(X2)) | → | pair#(X1, X2) | | tail#(mark(X)) | → | tail#(X) |
| take#(X1, mark(X2)) | → | take#(X1, X2) | | proper#(pair(X1, X2)) | → | proper#(X1) |
| proper#(s(X)) | → | proper#(X) | | active#(zip(X1, X2)) | → | active#(X2) |
| proper#(take(X1, X2)) | → | take#(proper(X1), proper(X2)) | | active#(zip(cons(X, XS), cons(Y, YS))) | → | cons#(pair(X, Y), zip(XS, YS)) |
| active#(incr(cons(X, XS))) | → | incr#(XS) | | active#(cons(X1, X2)) | → | active#(X1) |
| pair#(mark(X1), X2) | → | pair#(X1, X2) | | take#(mark(X1), X2) | → | take#(X1, X2) |
| cons#(mark(X1), X2) | → | cons#(X1, X2) | | zip#(mark(X1), X2) | → | zip#(X1, X2) |
| active#(pair(X1, X2)) | → | pair#(X1, active(X2)) | | repItems#(mark(X)) | → | repItems#(X) |
| proper#(incr(X)) | → | incr#(proper(X)) | | top#(ok(X)) | → | active#(X) |
| active#(zip(cons(X, XS), cons(Y, YS))) | → | zip#(XS, YS) | | active#(repItems(cons(X, XS))) | → | repItems#(XS) |
| active#(pair(X1, X2)) | → | active#(X1) | | proper#(take(X1, X2)) | → | proper#(X1) |
| repItems#(ok(X)) | → | repItems#(X) | | zip#(X1, mark(X2)) | → | zip#(X1, X2) |
| incr#(mark(X)) | → | incr#(X) | | proper#(zip(X1, X2)) | → | proper#(X1) |
| active#(take(s(N), cons(X, XS))) | → | take#(N, XS) | | take#(ok(X1), ok(X2)) | → | take#(X1, X2) |
| active#(repItems(X)) | → | active#(X) | | active#(incr(X)) | → | active#(X) |
| pair#(ok(X1), ok(X2)) | → | pair#(X1, X2) | | active#(incr(X)) | → | incr#(active(X)) |
| active#(take(X1, X2)) | → | active#(X1) | | proper#(take(X1, X2)) | → | proper#(X2) |
| active#(s(X)) | → | s#(active(X)) | | proper#(repItems(X)) | → | proper#(X) |
| s#(ok(X)) | → | s#(X) | | active#(incr(cons(X, XS))) | → | s#(X) |
| proper#(pair(X1, X2)) | → | pair#(proper(X1), proper(X2)) | | s#(mark(X)) | → | s#(X) |
| proper#(tail(X)) | → | tail#(proper(X)) | | proper#(cons(X1, X2)) | → | cons#(proper(X1), proper(X2)) |
| proper#(repItems(X)) | → | repItems#(proper(X)) | | active#(repItems(cons(X, XS))) | → | cons#(X, cons(X, repItems(XS))) |
| active#(s(X)) | → | active#(X) | | active#(take(X1, X2)) | → | take#(X1, active(X2)) |
| proper#(s(X)) | → | s#(proper(X)) | | active#(tail(X)) | → | active#(X) |
| active#(oddNs) | → | incr#(pairNs) | | active#(zip(X1, X2)) | → | zip#(active(X1), X2) |
| proper#(zip(X1, X2)) | → | proper#(X2) |
Rewrite Rules
| active(pairNs) | → | mark(cons(0, incr(oddNs))) | | active(oddNs) | → | mark(incr(pairNs)) |
| active(incr(cons(X, XS))) | → | mark(cons(s(X), incr(XS))) | | active(take(0, XS)) | → | mark(nil) |
| active(take(s(N), cons(X, XS))) | → | mark(cons(X, take(N, XS))) | | active(zip(nil, XS)) | → | mark(nil) |
| active(zip(X, nil)) | → | mark(nil) | | active(zip(cons(X, XS), cons(Y, YS))) | → | mark(cons(pair(X, Y), zip(XS, YS))) |
| active(tail(cons(X, XS))) | → | mark(XS) | | active(repItems(nil)) | → | mark(nil) |
| active(repItems(cons(X, XS))) | → | mark(cons(X, cons(X, repItems(XS)))) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(incr(X)) | → | incr(active(X)) | | active(s(X)) | → | s(active(X)) |
| active(take(X1, X2)) | → | take(active(X1), X2) | | active(take(X1, X2)) | → | take(X1, active(X2)) |
| active(zip(X1, X2)) | → | zip(active(X1), X2) | | active(zip(X1, X2)) | → | zip(X1, active(X2)) |
| active(pair(X1, X2)) | → | pair(active(X1), X2) | | active(pair(X1, X2)) | → | pair(X1, active(X2)) |
| active(tail(X)) | → | tail(active(X)) | | active(repItems(X)) | → | repItems(active(X)) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | incr(mark(X)) | → | mark(incr(X)) |
| s(mark(X)) | → | mark(s(X)) | | take(mark(X1), X2) | → | mark(take(X1, X2)) |
| take(X1, mark(X2)) | → | mark(take(X1, X2)) | | zip(mark(X1), X2) | → | mark(zip(X1, X2)) |
| zip(X1, mark(X2)) | → | mark(zip(X1, X2)) | | pair(mark(X1), X2) | → | mark(pair(X1, X2)) |
| pair(X1, mark(X2)) | → | mark(pair(X1, X2)) | | tail(mark(X)) | → | mark(tail(X)) |
| repItems(mark(X)) | → | mark(repItems(X)) | | proper(pairNs) | → | ok(pairNs) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(incr(X)) | → | incr(proper(X)) | | proper(oddNs) | → | ok(oddNs) |
| proper(s(X)) | → | s(proper(X)) | | proper(take(X1, X2)) | → | take(proper(X1), proper(X2)) |
| proper(nil) | → | ok(nil) | | proper(zip(X1, X2)) | → | zip(proper(X1), proper(X2)) |
| proper(pair(X1, X2)) | → | pair(proper(X1), proper(X2)) | | proper(tail(X)) | → | tail(proper(X)) |
| proper(repItems(X)) | → | repItems(proper(X)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| incr(ok(X)) | → | ok(incr(X)) | | s(ok(X)) | → | ok(s(X)) |
| take(ok(X1), ok(X2)) | → | ok(take(X1, X2)) | | zip(ok(X1), ok(X2)) | → | ok(zip(X1, X2)) |
| pair(ok(X1), ok(X2)) | → | ok(pair(X1, X2)) | | tail(ok(X)) | → | ok(tail(X)) |
| repItems(ok(X)) | → | ok(repItems(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: zip, pair, mark, tail, 0, pairNs, s, repItems, take, active, ok, proper, incr, oddNs, nil, cons, top
Strategy
The following SCCs where found
| cons#(mark(X1), X2) → cons#(X1, X2) | cons#(ok(X1), ok(X2)) → cons#(X1, X2) |
| zip#(mark(X1), X2) → zip#(X1, X2) | zip#(X1, mark(X2)) → zip#(X1, X2) |
| zip#(ok(X1), ok(X2)) → zip#(X1, X2) |
| tail#(ok(X)) → tail#(X) | tail#(mark(X)) → tail#(X) |
| proper#(s(X)) → proper#(X) | proper#(cons(X1, X2)) → proper#(X1) |
| proper#(cons(X1, X2)) → proper#(X2) | proper#(incr(X)) → proper#(X) |
| proper#(tail(X)) → proper#(X) | proper#(take(X1, X2)) → proper#(X1) |
| proper#(pair(X1, X2)) → proper#(X2) | proper#(take(X1, X2)) → proper#(X2) |
| proper#(zip(X1, X2)) → proper#(X1) | proper#(repItems(X)) → proper#(X) |
| proper#(pair(X1, X2)) → proper#(X1) | proper#(zip(X1, X2)) → proper#(X2) |
| pair#(mark(X1), X2) → pair#(X1, X2) | pair#(ok(X1), ok(X2)) → pair#(X1, X2) |
| pair#(X1, mark(X2)) → pair#(X1, X2) |
| incr#(ok(X)) → incr#(X) | incr#(mark(X)) → incr#(X) |
| repItems#(ok(X)) → repItems#(X) | repItems#(mark(X)) → repItems#(X) |
| take#(mark(X1), X2) → take#(X1, X2) | take#(X1, mark(X2)) → take#(X1, X2) |
| take#(ok(X1), ok(X2)) → take#(X1, X2) |
| active#(repItems(X)) → active#(X) | active#(zip(X1, X2)) → active#(X2) |
| active#(incr(X)) → active#(X) | active#(pair(X1, X2)) → active#(X2) |
| active#(s(X)) → active#(X) | active#(pair(X1, X2)) → active#(X1) |
| active#(take(X1, X2)) → active#(X2) | active#(tail(X)) → active#(X) |
| active#(take(X1, X2)) → active#(X1) | active#(zip(X1, X2)) → active#(X1) |
| active#(cons(X1, X2)) → active#(X1) |
| top#(mark(X)) → top#(proper(X)) | top#(ok(X)) → top#(active(X)) |
| s#(mark(X)) → s#(X) | s#(ok(X)) → s#(X) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| incr#(ok(X)) | → | incr#(X) | | incr#(mark(X)) | → | incr#(X) |
Rewrite Rules
| active(pairNs) | → | mark(cons(0, incr(oddNs))) | | active(oddNs) | → | mark(incr(pairNs)) |
| active(incr(cons(X, XS))) | → | mark(cons(s(X), incr(XS))) | | active(take(0, XS)) | → | mark(nil) |
| active(take(s(N), cons(X, XS))) | → | mark(cons(X, take(N, XS))) | | active(zip(nil, XS)) | → | mark(nil) |
| active(zip(X, nil)) | → | mark(nil) | | active(zip(cons(X, XS), cons(Y, YS))) | → | mark(cons(pair(X, Y), zip(XS, YS))) |
| active(tail(cons(X, XS))) | → | mark(XS) | | active(repItems(nil)) | → | mark(nil) |
| active(repItems(cons(X, XS))) | → | mark(cons(X, cons(X, repItems(XS)))) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(incr(X)) | → | incr(active(X)) | | active(s(X)) | → | s(active(X)) |
| active(take(X1, X2)) | → | take(active(X1), X2) | | active(take(X1, X2)) | → | take(X1, active(X2)) |
| active(zip(X1, X2)) | → | zip(active(X1), X2) | | active(zip(X1, X2)) | → | zip(X1, active(X2)) |
| active(pair(X1, X2)) | → | pair(active(X1), X2) | | active(pair(X1, X2)) | → | pair(X1, active(X2)) |
| active(tail(X)) | → | tail(active(X)) | | active(repItems(X)) | → | repItems(active(X)) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | incr(mark(X)) | → | mark(incr(X)) |
| s(mark(X)) | → | mark(s(X)) | | take(mark(X1), X2) | → | mark(take(X1, X2)) |
| take(X1, mark(X2)) | → | mark(take(X1, X2)) | | zip(mark(X1), X2) | → | mark(zip(X1, X2)) |
| zip(X1, mark(X2)) | → | mark(zip(X1, X2)) | | pair(mark(X1), X2) | → | mark(pair(X1, X2)) |
| pair(X1, mark(X2)) | → | mark(pair(X1, X2)) | | tail(mark(X)) | → | mark(tail(X)) |
| repItems(mark(X)) | → | mark(repItems(X)) | | proper(pairNs) | → | ok(pairNs) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(incr(X)) | → | incr(proper(X)) | | proper(oddNs) | → | ok(oddNs) |
| proper(s(X)) | → | s(proper(X)) | | proper(take(X1, X2)) | → | take(proper(X1), proper(X2)) |
| proper(nil) | → | ok(nil) | | proper(zip(X1, X2)) | → | zip(proper(X1), proper(X2)) |
| proper(pair(X1, X2)) | → | pair(proper(X1), proper(X2)) | | proper(tail(X)) | → | tail(proper(X)) |
| proper(repItems(X)) | → | repItems(proper(X)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| incr(ok(X)) | → | ok(incr(X)) | | s(ok(X)) | → | ok(s(X)) |
| take(ok(X1), ok(X2)) | → | ok(take(X1, X2)) | | zip(ok(X1), ok(X2)) | → | ok(zip(X1, X2)) |
| pair(ok(X1), ok(X2)) | → | ok(pair(X1, X2)) | | tail(ok(X)) | → | ok(tail(X)) |
| repItems(ok(X)) | → | ok(repItems(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: zip, pair, mark, tail, 0, pairNs, s, repItems, take, active, ok, proper, incr, oddNs, nil, cons, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| incr#(ok(X)) | → | incr#(X) | | incr#(mark(X)) | → | incr#(X) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| cons#(mark(X1), X2) | → | cons#(X1, X2) | | cons#(ok(X1), ok(X2)) | → | cons#(X1, X2) |
Rewrite Rules
| active(pairNs) | → | mark(cons(0, incr(oddNs))) | | active(oddNs) | → | mark(incr(pairNs)) |
| active(incr(cons(X, XS))) | → | mark(cons(s(X), incr(XS))) | | active(take(0, XS)) | → | mark(nil) |
| active(take(s(N), cons(X, XS))) | → | mark(cons(X, take(N, XS))) | | active(zip(nil, XS)) | → | mark(nil) |
| active(zip(X, nil)) | → | mark(nil) | | active(zip(cons(X, XS), cons(Y, YS))) | → | mark(cons(pair(X, Y), zip(XS, YS))) |
| active(tail(cons(X, XS))) | → | mark(XS) | | active(repItems(nil)) | → | mark(nil) |
| active(repItems(cons(X, XS))) | → | mark(cons(X, cons(X, repItems(XS)))) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(incr(X)) | → | incr(active(X)) | | active(s(X)) | → | s(active(X)) |
| active(take(X1, X2)) | → | take(active(X1), X2) | | active(take(X1, X2)) | → | take(X1, active(X2)) |
| active(zip(X1, X2)) | → | zip(active(X1), X2) | | active(zip(X1, X2)) | → | zip(X1, active(X2)) |
| active(pair(X1, X2)) | → | pair(active(X1), X2) | | active(pair(X1, X2)) | → | pair(X1, active(X2)) |
| active(tail(X)) | → | tail(active(X)) | | active(repItems(X)) | → | repItems(active(X)) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | incr(mark(X)) | → | mark(incr(X)) |
| s(mark(X)) | → | mark(s(X)) | | take(mark(X1), X2) | → | mark(take(X1, X2)) |
| take(X1, mark(X2)) | → | mark(take(X1, X2)) | | zip(mark(X1), X2) | → | mark(zip(X1, X2)) |
| zip(X1, mark(X2)) | → | mark(zip(X1, X2)) | | pair(mark(X1), X2) | → | mark(pair(X1, X2)) |
| pair(X1, mark(X2)) | → | mark(pair(X1, X2)) | | tail(mark(X)) | → | mark(tail(X)) |
| repItems(mark(X)) | → | mark(repItems(X)) | | proper(pairNs) | → | ok(pairNs) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(incr(X)) | → | incr(proper(X)) | | proper(oddNs) | → | ok(oddNs) |
| proper(s(X)) | → | s(proper(X)) | | proper(take(X1, X2)) | → | take(proper(X1), proper(X2)) |
| proper(nil) | → | ok(nil) | | proper(zip(X1, X2)) | → | zip(proper(X1), proper(X2)) |
| proper(pair(X1, X2)) | → | pair(proper(X1), proper(X2)) | | proper(tail(X)) | → | tail(proper(X)) |
| proper(repItems(X)) | → | repItems(proper(X)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| incr(ok(X)) | → | ok(incr(X)) | | s(ok(X)) | → | ok(s(X)) |
| take(ok(X1), ok(X2)) | → | ok(take(X1, X2)) | | zip(ok(X1), ok(X2)) | → | ok(zip(X1, X2)) |
| pair(ok(X1), ok(X2)) | → | ok(pair(X1, X2)) | | tail(ok(X)) | → | ok(tail(X)) |
| repItems(ok(X)) | → | ok(repItems(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: zip, pair, mark, tail, 0, pairNs, s, repItems, take, active, ok, proper, incr, oddNs, nil, cons, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| cons#(mark(X1), X2) | → | cons#(X1, X2) | | cons#(ok(X1), ok(X2)) | → | cons#(X1, X2) |
Problem 4: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| pair#(mark(X1), X2) | → | pair#(X1, X2) | | pair#(ok(X1), ok(X2)) | → | pair#(X1, X2) |
| pair#(X1, mark(X2)) | → | pair#(X1, X2) |
Rewrite Rules
| active(pairNs) | → | mark(cons(0, incr(oddNs))) | | active(oddNs) | → | mark(incr(pairNs)) |
| active(incr(cons(X, XS))) | → | mark(cons(s(X), incr(XS))) | | active(take(0, XS)) | → | mark(nil) |
| active(take(s(N), cons(X, XS))) | → | mark(cons(X, take(N, XS))) | | active(zip(nil, XS)) | → | mark(nil) |
| active(zip(X, nil)) | → | mark(nil) | | active(zip(cons(X, XS), cons(Y, YS))) | → | mark(cons(pair(X, Y), zip(XS, YS))) |
| active(tail(cons(X, XS))) | → | mark(XS) | | active(repItems(nil)) | → | mark(nil) |
| active(repItems(cons(X, XS))) | → | mark(cons(X, cons(X, repItems(XS)))) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(incr(X)) | → | incr(active(X)) | | active(s(X)) | → | s(active(X)) |
| active(take(X1, X2)) | → | take(active(X1), X2) | | active(take(X1, X2)) | → | take(X1, active(X2)) |
| active(zip(X1, X2)) | → | zip(active(X1), X2) | | active(zip(X1, X2)) | → | zip(X1, active(X2)) |
| active(pair(X1, X2)) | → | pair(active(X1), X2) | | active(pair(X1, X2)) | → | pair(X1, active(X2)) |
| active(tail(X)) | → | tail(active(X)) | | active(repItems(X)) | → | repItems(active(X)) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | incr(mark(X)) | → | mark(incr(X)) |
| s(mark(X)) | → | mark(s(X)) | | take(mark(X1), X2) | → | mark(take(X1, X2)) |
| take(X1, mark(X2)) | → | mark(take(X1, X2)) | | zip(mark(X1), X2) | → | mark(zip(X1, X2)) |
| zip(X1, mark(X2)) | → | mark(zip(X1, X2)) | | pair(mark(X1), X2) | → | mark(pair(X1, X2)) |
| pair(X1, mark(X2)) | → | mark(pair(X1, X2)) | | tail(mark(X)) | → | mark(tail(X)) |
| repItems(mark(X)) | → | mark(repItems(X)) | | proper(pairNs) | → | ok(pairNs) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(incr(X)) | → | incr(proper(X)) | | proper(oddNs) | → | ok(oddNs) |
| proper(s(X)) | → | s(proper(X)) | | proper(take(X1, X2)) | → | take(proper(X1), proper(X2)) |
| proper(nil) | → | ok(nil) | | proper(zip(X1, X2)) | → | zip(proper(X1), proper(X2)) |
| proper(pair(X1, X2)) | → | pair(proper(X1), proper(X2)) | | proper(tail(X)) | → | tail(proper(X)) |
| proper(repItems(X)) | → | repItems(proper(X)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| incr(ok(X)) | → | ok(incr(X)) | | s(ok(X)) | → | ok(s(X)) |
| take(ok(X1), ok(X2)) | → | ok(take(X1, X2)) | | zip(ok(X1), ok(X2)) | → | ok(zip(X1, X2)) |
| pair(ok(X1), ok(X2)) | → | ok(pair(X1, X2)) | | tail(ok(X)) | → | ok(tail(X)) |
| repItems(ok(X)) | → | ok(repItems(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: zip, pair, mark, tail, 0, pairNs, s, repItems, take, active, ok, proper, incr, oddNs, nil, cons, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| pair#(mark(X1), X2) | → | pair#(X1, X2) | | pair#(ok(X1), ok(X2)) | → | pair#(X1, X2) |
Problem 13: ReductionPairSAT
Dependency Pair Problem
Dependency Pairs
| pair#(X1, mark(X2)) | → | pair#(X1, X2) |
Rewrite Rules
| active(pairNs) | → | mark(cons(0, incr(oddNs))) | | active(oddNs) | → | mark(incr(pairNs)) |
| active(incr(cons(X, XS))) | → | mark(cons(s(X), incr(XS))) | | active(take(0, XS)) | → | mark(nil) |
| active(take(s(N), cons(X, XS))) | → | mark(cons(X, take(N, XS))) | | active(zip(nil, XS)) | → | mark(nil) |
| active(zip(X, nil)) | → | mark(nil) | | active(zip(cons(X, XS), cons(Y, YS))) | → | mark(cons(pair(X, Y), zip(XS, YS))) |
| active(tail(cons(X, XS))) | → | mark(XS) | | active(repItems(nil)) | → | mark(nil) |
| active(repItems(cons(X, XS))) | → | mark(cons(X, cons(X, repItems(XS)))) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(incr(X)) | → | incr(active(X)) | | active(s(X)) | → | s(active(X)) |
| active(take(X1, X2)) | → | take(active(X1), X2) | | active(take(X1, X2)) | → | take(X1, active(X2)) |
| active(zip(X1, X2)) | → | zip(active(X1), X2) | | active(zip(X1, X2)) | → | zip(X1, active(X2)) |
| active(pair(X1, X2)) | → | pair(active(X1), X2) | | active(pair(X1, X2)) | → | pair(X1, active(X2)) |
| active(tail(X)) | → | tail(active(X)) | | active(repItems(X)) | → | repItems(active(X)) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | incr(mark(X)) | → | mark(incr(X)) |
| s(mark(X)) | → | mark(s(X)) | | take(mark(X1), X2) | → | mark(take(X1, X2)) |
| take(X1, mark(X2)) | → | mark(take(X1, X2)) | | zip(mark(X1), X2) | → | mark(zip(X1, X2)) |
| zip(X1, mark(X2)) | → | mark(zip(X1, X2)) | | pair(mark(X1), X2) | → | mark(pair(X1, X2)) |
| pair(X1, mark(X2)) | → | mark(pair(X1, X2)) | | tail(mark(X)) | → | mark(tail(X)) |
| repItems(mark(X)) | → | mark(repItems(X)) | | proper(pairNs) | → | ok(pairNs) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(incr(X)) | → | incr(proper(X)) | | proper(oddNs) | → | ok(oddNs) |
| proper(s(X)) | → | s(proper(X)) | | proper(take(X1, X2)) | → | take(proper(X1), proper(X2)) |
| proper(nil) | → | ok(nil) | | proper(zip(X1, X2)) | → | zip(proper(X1), proper(X2)) |
| proper(pair(X1, X2)) | → | pair(proper(X1), proper(X2)) | | proper(tail(X)) | → | tail(proper(X)) |
| proper(repItems(X)) | → | repItems(proper(X)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| incr(ok(X)) | → | ok(incr(X)) | | s(ok(X)) | → | ok(s(X)) |
| take(ok(X1), ok(X2)) | → | ok(take(X1, X2)) | | zip(ok(X1), ok(X2)) | → | ok(zip(X1, X2)) |
| pair(ok(X1), ok(X2)) | → | ok(pair(X1, X2)) | | tail(ok(X)) | → | ok(tail(X)) |
| repItems(ok(X)) | → | ok(repItems(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: zip, pair, mark, tail, 0, pairNs, s, repItems, take, active, ok, proper, incr, oddNs, top, cons, nil
Strategy
Function Precedence
zip = pair = mark = tail = pair# = 0 = s = pairNs = repItems = take = active = ok = proper = incr = oddNs = top = cons = nil
Argument Filtering
zip: all arguments are removed from zip
pair: 1 2
mark: 1
tail: all arguments are removed from tail
pair#: 2
0: all arguments are removed from 0
s: all arguments are removed from s
pairNs: all arguments are removed from pairNs
repItems: collapses to 1
take: 1 2
active: all arguments are removed from active
ok: all arguments are removed from ok
proper: all arguments are removed from proper
incr: all arguments are removed from incr
oddNs: all arguments are removed from oddNs
top: 1
cons: all arguments are removed from cons
nil: all arguments are removed from nil
Status
zip: multiset
pair: lexicographic with permutation 1 → 1 2 → 2
mark: multiset
tail: multiset
pair#: lexicographic with permutation 2 → 1
0: multiset
s: multiset
pairNs: multiset
take: lexicographic with permutation 1 → 1 2 → 2
active: multiset
ok: multiset
proper: multiset
incr: multiset
oddNs: multiset
top: lexicographic with permutation 1 → 1
cons: multiset
nil: 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.
| pair#(X1, mark(X2)) → pair#(X1, X2) |
Problem 5: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| proper#(s(X)) | → | proper#(X) | | proper#(cons(X1, X2)) | → | proper#(X1) |
| proper#(cons(X1, X2)) | → | proper#(X2) | | proper#(incr(X)) | → | proper#(X) |
| proper#(tail(X)) | → | proper#(X) | | proper#(take(X1, X2)) | → | proper#(X1) |
| proper#(pair(X1, X2)) | → | proper#(X2) | | proper#(take(X1, X2)) | → | proper#(X2) |
| proper#(zip(X1, X2)) | → | proper#(X1) | | proper#(repItems(X)) | → | proper#(X) |
| proper#(zip(X1, X2)) | → | proper#(X2) | | proper#(pair(X1, X2)) | → | proper#(X1) |
Rewrite Rules
| active(pairNs) | → | mark(cons(0, incr(oddNs))) | | active(oddNs) | → | mark(incr(pairNs)) |
| active(incr(cons(X, XS))) | → | mark(cons(s(X), incr(XS))) | | active(take(0, XS)) | → | mark(nil) |
| active(take(s(N), cons(X, XS))) | → | mark(cons(X, take(N, XS))) | | active(zip(nil, XS)) | → | mark(nil) |
| active(zip(X, nil)) | → | mark(nil) | | active(zip(cons(X, XS), cons(Y, YS))) | → | mark(cons(pair(X, Y), zip(XS, YS))) |
| active(tail(cons(X, XS))) | → | mark(XS) | | active(repItems(nil)) | → | mark(nil) |
| active(repItems(cons(X, XS))) | → | mark(cons(X, cons(X, repItems(XS)))) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(incr(X)) | → | incr(active(X)) | | active(s(X)) | → | s(active(X)) |
| active(take(X1, X2)) | → | take(active(X1), X2) | | active(take(X1, X2)) | → | take(X1, active(X2)) |
| active(zip(X1, X2)) | → | zip(active(X1), X2) | | active(zip(X1, X2)) | → | zip(X1, active(X2)) |
| active(pair(X1, X2)) | → | pair(active(X1), X2) | | active(pair(X1, X2)) | → | pair(X1, active(X2)) |
| active(tail(X)) | → | tail(active(X)) | | active(repItems(X)) | → | repItems(active(X)) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | incr(mark(X)) | → | mark(incr(X)) |
| s(mark(X)) | → | mark(s(X)) | | take(mark(X1), X2) | → | mark(take(X1, X2)) |
| take(X1, mark(X2)) | → | mark(take(X1, X2)) | | zip(mark(X1), X2) | → | mark(zip(X1, X2)) |
| zip(X1, mark(X2)) | → | mark(zip(X1, X2)) | | pair(mark(X1), X2) | → | mark(pair(X1, X2)) |
| pair(X1, mark(X2)) | → | mark(pair(X1, X2)) | | tail(mark(X)) | → | mark(tail(X)) |
| repItems(mark(X)) | → | mark(repItems(X)) | | proper(pairNs) | → | ok(pairNs) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(incr(X)) | → | incr(proper(X)) | | proper(oddNs) | → | ok(oddNs) |
| proper(s(X)) | → | s(proper(X)) | | proper(take(X1, X2)) | → | take(proper(X1), proper(X2)) |
| proper(nil) | → | ok(nil) | | proper(zip(X1, X2)) | → | zip(proper(X1), proper(X2)) |
| proper(pair(X1, X2)) | → | pair(proper(X1), proper(X2)) | | proper(tail(X)) | → | tail(proper(X)) |
| proper(repItems(X)) | → | repItems(proper(X)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| incr(ok(X)) | → | ok(incr(X)) | | s(ok(X)) | → | ok(s(X)) |
| take(ok(X1), ok(X2)) | → | ok(take(X1, X2)) | | zip(ok(X1), ok(X2)) | → | ok(zip(X1, X2)) |
| pair(ok(X1), ok(X2)) | → | ok(pair(X1, X2)) | | tail(ok(X)) | → | ok(tail(X)) |
| repItems(ok(X)) | → | ok(repItems(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: zip, pair, mark, tail, 0, pairNs, s, repItems, take, active, ok, proper, incr, oddNs, nil, cons, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| proper#(s(X)) | → | proper#(X) | | proper#(cons(X1, X2)) | → | proper#(X1) |
| proper#(cons(X1, X2)) | → | proper#(X2) | | proper#(tail(X)) | → | proper#(X) |
| proper#(incr(X)) | → | proper#(X) | | proper#(take(X1, X2)) | → | proper#(X1) |
| proper#(pair(X1, X2)) | → | proper#(X2) | | proper#(take(X1, X2)) | → | proper#(X2) |
| proper#(zip(X1, X2)) | → | proper#(X1) | | proper#(repItems(X)) | → | proper#(X) |
| proper#(pair(X1, X2)) | → | proper#(X1) | | proper#(zip(X1, X2)) | → | proper#(X2) |
Problem 6: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| s#(mark(X)) | → | s#(X) | | s#(ok(X)) | → | s#(X) |
Rewrite Rules
| active(pairNs) | → | mark(cons(0, incr(oddNs))) | | active(oddNs) | → | mark(incr(pairNs)) |
| active(incr(cons(X, XS))) | → | mark(cons(s(X), incr(XS))) | | active(take(0, XS)) | → | mark(nil) |
| active(take(s(N), cons(X, XS))) | → | mark(cons(X, take(N, XS))) | | active(zip(nil, XS)) | → | mark(nil) |
| active(zip(X, nil)) | → | mark(nil) | | active(zip(cons(X, XS), cons(Y, YS))) | → | mark(cons(pair(X, Y), zip(XS, YS))) |
| active(tail(cons(X, XS))) | → | mark(XS) | | active(repItems(nil)) | → | mark(nil) |
| active(repItems(cons(X, XS))) | → | mark(cons(X, cons(X, repItems(XS)))) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(incr(X)) | → | incr(active(X)) | | active(s(X)) | → | s(active(X)) |
| active(take(X1, X2)) | → | take(active(X1), X2) | | active(take(X1, X2)) | → | take(X1, active(X2)) |
| active(zip(X1, X2)) | → | zip(active(X1), X2) | | active(zip(X1, X2)) | → | zip(X1, active(X2)) |
| active(pair(X1, X2)) | → | pair(active(X1), X2) | | active(pair(X1, X2)) | → | pair(X1, active(X2)) |
| active(tail(X)) | → | tail(active(X)) | | active(repItems(X)) | → | repItems(active(X)) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | incr(mark(X)) | → | mark(incr(X)) |
| s(mark(X)) | → | mark(s(X)) | | take(mark(X1), X2) | → | mark(take(X1, X2)) |
| take(X1, mark(X2)) | → | mark(take(X1, X2)) | | zip(mark(X1), X2) | → | mark(zip(X1, X2)) |
| zip(X1, mark(X2)) | → | mark(zip(X1, X2)) | | pair(mark(X1), X2) | → | mark(pair(X1, X2)) |
| pair(X1, mark(X2)) | → | mark(pair(X1, X2)) | | tail(mark(X)) | → | mark(tail(X)) |
| repItems(mark(X)) | → | mark(repItems(X)) | | proper(pairNs) | → | ok(pairNs) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(incr(X)) | → | incr(proper(X)) | | proper(oddNs) | → | ok(oddNs) |
| proper(s(X)) | → | s(proper(X)) | | proper(take(X1, X2)) | → | take(proper(X1), proper(X2)) |
| proper(nil) | → | ok(nil) | | proper(zip(X1, X2)) | → | zip(proper(X1), proper(X2)) |
| proper(pair(X1, X2)) | → | pair(proper(X1), proper(X2)) | | proper(tail(X)) | → | tail(proper(X)) |
| proper(repItems(X)) | → | repItems(proper(X)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| incr(ok(X)) | → | ok(incr(X)) | | s(ok(X)) | → | ok(s(X)) |
| take(ok(X1), ok(X2)) | → | ok(take(X1, X2)) | | zip(ok(X1), ok(X2)) | → | ok(zip(X1, X2)) |
| pair(ok(X1), ok(X2)) | → | ok(pair(X1, X2)) | | tail(ok(X)) | → | ok(tail(X)) |
| repItems(ok(X)) | → | ok(repItems(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: zip, pair, mark, tail, 0, pairNs, s, repItems, take, active, ok, proper, incr, oddNs, nil, cons, top
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 7: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| tail#(ok(X)) | → | tail#(X) | | tail#(mark(X)) | → | tail#(X) |
Rewrite Rules
| active(pairNs) | → | mark(cons(0, incr(oddNs))) | | active(oddNs) | → | mark(incr(pairNs)) |
| active(incr(cons(X, XS))) | → | mark(cons(s(X), incr(XS))) | | active(take(0, XS)) | → | mark(nil) |
| active(take(s(N), cons(X, XS))) | → | mark(cons(X, take(N, XS))) | | active(zip(nil, XS)) | → | mark(nil) |
| active(zip(X, nil)) | → | mark(nil) | | active(zip(cons(X, XS), cons(Y, YS))) | → | mark(cons(pair(X, Y), zip(XS, YS))) |
| active(tail(cons(X, XS))) | → | mark(XS) | | active(repItems(nil)) | → | mark(nil) |
| active(repItems(cons(X, XS))) | → | mark(cons(X, cons(X, repItems(XS)))) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(incr(X)) | → | incr(active(X)) | | active(s(X)) | → | s(active(X)) |
| active(take(X1, X2)) | → | take(active(X1), X2) | | active(take(X1, X2)) | → | take(X1, active(X2)) |
| active(zip(X1, X2)) | → | zip(active(X1), X2) | | active(zip(X1, X2)) | → | zip(X1, active(X2)) |
| active(pair(X1, X2)) | → | pair(active(X1), X2) | | active(pair(X1, X2)) | → | pair(X1, active(X2)) |
| active(tail(X)) | → | tail(active(X)) | | active(repItems(X)) | → | repItems(active(X)) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | incr(mark(X)) | → | mark(incr(X)) |
| s(mark(X)) | → | mark(s(X)) | | take(mark(X1), X2) | → | mark(take(X1, X2)) |
| take(X1, mark(X2)) | → | mark(take(X1, X2)) | | zip(mark(X1), X2) | → | mark(zip(X1, X2)) |
| zip(X1, mark(X2)) | → | mark(zip(X1, X2)) | | pair(mark(X1), X2) | → | mark(pair(X1, X2)) |
| pair(X1, mark(X2)) | → | mark(pair(X1, X2)) | | tail(mark(X)) | → | mark(tail(X)) |
| repItems(mark(X)) | → | mark(repItems(X)) | | proper(pairNs) | → | ok(pairNs) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(incr(X)) | → | incr(proper(X)) | | proper(oddNs) | → | ok(oddNs) |
| proper(s(X)) | → | s(proper(X)) | | proper(take(X1, X2)) | → | take(proper(X1), proper(X2)) |
| proper(nil) | → | ok(nil) | | proper(zip(X1, X2)) | → | zip(proper(X1), proper(X2)) |
| proper(pair(X1, X2)) | → | pair(proper(X1), proper(X2)) | | proper(tail(X)) | → | tail(proper(X)) |
| proper(repItems(X)) | → | repItems(proper(X)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| incr(ok(X)) | → | ok(incr(X)) | | s(ok(X)) | → | ok(s(X)) |
| take(ok(X1), ok(X2)) | → | ok(take(X1, X2)) | | zip(ok(X1), ok(X2)) | → | ok(zip(X1, X2)) |
| pair(ok(X1), ok(X2)) | → | ok(pair(X1, X2)) | | tail(ok(X)) | → | ok(tail(X)) |
| repItems(ok(X)) | → | ok(repItems(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: zip, pair, mark, tail, 0, pairNs, s, repItems, take, active, ok, proper, incr, oddNs, nil, cons, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| tail#(ok(X)) | → | tail#(X) | | tail#(mark(X)) | → | tail#(X) |
Problem 8: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| zip#(mark(X1), X2) | → | zip#(X1, X2) | | zip#(X1, mark(X2)) | → | zip#(X1, X2) |
| zip#(ok(X1), ok(X2)) | → | zip#(X1, X2) |
Rewrite Rules
| active(pairNs) | → | mark(cons(0, incr(oddNs))) | | active(oddNs) | → | mark(incr(pairNs)) |
| active(incr(cons(X, XS))) | → | mark(cons(s(X), incr(XS))) | | active(take(0, XS)) | → | mark(nil) |
| active(take(s(N), cons(X, XS))) | → | mark(cons(X, take(N, XS))) | | active(zip(nil, XS)) | → | mark(nil) |
| active(zip(X, nil)) | → | mark(nil) | | active(zip(cons(X, XS), cons(Y, YS))) | → | mark(cons(pair(X, Y), zip(XS, YS))) |
| active(tail(cons(X, XS))) | → | mark(XS) | | active(repItems(nil)) | → | mark(nil) |
| active(repItems(cons(X, XS))) | → | mark(cons(X, cons(X, repItems(XS)))) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(incr(X)) | → | incr(active(X)) | | active(s(X)) | → | s(active(X)) |
| active(take(X1, X2)) | → | take(active(X1), X2) | | active(take(X1, X2)) | → | take(X1, active(X2)) |
| active(zip(X1, X2)) | → | zip(active(X1), X2) | | active(zip(X1, X2)) | → | zip(X1, active(X2)) |
| active(pair(X1, X2)) | → | pair(active(X1), X2) | | active(pair(X1, X2)) | → | pair(X1, active(X2)) |
| active(tail(X)) | → | tail(active(X)) | | active(repItems(X)) | → | repItems(active(X)) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | incr(mark(X)) | → | mark(incr(X)) |
| s(mark(X)) | → | mark(s(X)) | | take(mark(X1), X2) | → | mark(take(X1, X2)) |
| take(X1, mark(X2)) | → | mark(take(X1, X2)) | | zip(mark(X1), X2) | → | mark(zip(X1, X2)) |
| zip(X1, mark(X2)) | → | mark(zip(X1, X2)) | | pair(mark(X1), X2) | → | mark(pair(X1, X2)) |
| pair(X1, mark(X2)) | → | mark(pair(X1, X2)) | | tail(mark(X)) | → | mark(tail(X)) |
| repItems(mark(X)) | → | mark(repItems(X)) | | proper(pairNs) | → | ok(pairNs) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(incr(X)) | → | incr(proper(X)) | | proper(oddNs) | → | ok(oddNs) |
| proper(s(X)) | → | s(proper(X)) | | proper(take(X1, X2)) | → | take(proper(X1), proper(X2)) |
| proper(nil) | → | ok(nil) | | proper(zip(X1, X2)) | → | zip(proper(X1), proper(X2)) |
| proper(pair(X1, X2)) | → | pair(proper(X1), proper(X2)) | | proper(tail(X)) | → | tail(proper(X)) |
| proper(repItems(X)) | → | repItems(proper(X)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| incr(ok(X)) | → | ok(incr(X)) | | s(ok(X)) | → | ok(s(X)) |
| take(ok(X1), ok(X2)) | → | ok(take(X1, X2)) | | zip(ok(X1), ok(X2)) | → | ok(zip(X1, X2)) |
| pair(ok(X1), ok(X2)) | → | ok(pair(X1, X2)) | | tail(ok(X)) | → | ok(tail(X)) |
| repItems(ok(X)) | → | ok(repItems(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: zip, pair, mark, tail, 0, pairNs, s, repItems, take, active, ok, proper, incr, oddNs, nil, cons, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| zip#(mark(X1), X2) | → | zip#(X1, X2) | | zip#(ok(X1), ok(X2)) | → | zip#(X1, X2) |
Problem 14: ReductionPairSAT
Dependency Pair Problem
Dependency Pairs
| zip#(X1, mark(X2)) | → | zip#(X1, X2) |
Rewrite Rules
| active(pairNs) | → | mark(cons(0, incr(oddNs))) | | active(oddNs) | → | mark(incr(pairNs)) |
| active(incr(cons(X, XS))) | → | mark(cons(s(X), incr(XS))) | | active(take(0, XS)) | → | mark(nil) |
| active(take(s(N), cons(X, XS))) | → | mark(cons(X, take(N, XS))) | | active(zip(nil, XS)) | → | mark(nil) |
| active(zip(X, nil)) | → | mark(nil) | | active(zip(cons(X, XS), cons(Y, YS))) | → | mark(cons(pair(X, Y), zip(XS, YS))) |
| active(tail(cons(X, XS))) | → | mark(XS) | | active(repItems(nil)) | → | mark(nil) |
| active(repItems(cons(X, XS))) | → | mark(cons(X, cons(X, repItems(XS)))) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(incr(X)) | → | incr(active(X)) | | active(s(X)) | → | s(active(X)) |
| active(take(X1, X2)) | → | take(active(X1), X2) | | active(take(X1, X2)) | → | take(X1, active(X2)) |
| active(zip(X1, X2)) | → | zip(active(X1), X2) | | active(zip(X1, X2)) | → | zip(X1, active(X2)) |
| active(pair(X1, X2)) | → | pair(active(X1), X2) | | active(pair(X1, X2)) | → | pair(X1, active(X2)) |
| active(tail(X)) | → | tail(active(X)) | | active(repItems(X)) | → | repItems(active(X)) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | incr(mark(X)) | → | mark(incr(X)) |
| s(mark(X)) | → | mark(s(X)) | | take(mark(X1), X2) | → | mark(take(X1, X2)) |
| take(X1, mark(X2)) | → | mark(take(X1, X2)) | | zip(mark(X1), X2) | → | mark(zip(X1, X2)) |
| zip(X1, mark(X2)) | → | mark(zip(X1, X2)) | | pair(mark(X1), X2) | → | mark(pair(X1, X2)) |
| pair(X1, mark(X2)) | → | mark(pair(X1, X2)) | | tail(mark(X)) | → | mark(tail(X)) |
| repItems(mark(X)) | → | mark(repItems(X)) | | proper(pairNs) | → | ok(pairNs) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(incr(X)) | → | incr(proper(X)) | | proper(oddNs) | → | ok(oddNs) |
| proper(s(X)) | → | s(proper(X)) | | proper(take(X1, X2)) | → | take(proper(X1), proper(X2)) |
| proper(nil) | → | ok(nil) | | proper(zip(X1, X2)) | → | zip(proper(X1), proper(X2)) |
| proper(pair(X1, X2)) | → | pair(proper(X1), proper(X2)) | | proper(tail(X)) | → | tail(proper(X)) |
| proper(repItems(X)) | → | repItems(proper(X)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| incr(ok(X)) | → | ok(incr(X)) | | s(ok(X)) | → | ok(s(X)) |
| take(ok(X1), ok(X2)) | → | ok(take(X1, X2)) | | zip(ok(X1), ok(X2)) | → | ok(zip(X1, X2)) |
| pair(ok(X1), ok(X2)) | → | ok(pair(X1, X2)) | | tail(ok(X)) | → | ok(tail(X)) |
| repItems(ok(X)) | → | ok(repItems(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: zip, pair, mark, tail, 0, pairNs, s, repItems, take, active, ok, proper, incr, oddNs, top, cons, nil
Strategy
Function Precedence
zip# < mark < zip = pair = tail = 0 = s = pairNs = repItems = take = active = ok = proper = incr = oddNs = top = cons = nil
Argument Filtering
zip: all arguments are removed from zip
pair: all arguments are removed from pair
zip#: 2
mark: 1
tail: all arguments are removed from tail
0: all arguments are removed from 0
s: all arguments are removed from s
pairNs: all arguments are removed from pairNs
repItems: all arguments are removed from repItems
take: all arguments are removed from take
active: all arguments are removed from active
ok: all arguments are removed from ok
proper: all arguments are removed from proper
incr: all arguments are removed from incr
oddNs: all arguments are removed from oddNs
top: collapses to 1
cons: collapses to 1
nil: all arguments are removed from nil
Status
zip: multiset
pair: multiset
zip#: lexicographic with permutation 2 → 1
mark: multiset
tail: multiset
0: multiset
s: multiset
pairNs: multiset
repItems: multiset
take: multiset
active: multiset
ok: multiset
proper: multiset
incr: multiset
oddNs: multiset
nil: 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.
| zip#(X1, mark(X2)) → zip#(X1, X2) |
Problem 10: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| active#(repItems(X)) | → | active#(X) | | active#(zip(X1, X2)) | → | active#(X2) |
| active#(incr(X)) | → | active#(X) | | active#(pair(X1, X2)) | → | active#(X2) |
| active#(s(X)) | → | active#(X) | | active#(pair(X1, X2)) | → | active#(X1) |
| active#(take(X1, X2)) | → | active#(X2) | | active#(tail(X)) | → | active#(X) |
| active#(take(X1, X2)) | → | active#(X1) | | active#(zip(X1, X2)) | → | active#(X1) |
| active#(cons(X1, X2)) | → | active#(X1) |
Rewrite Rules
| active(pairNs) | → | mark(cons(0, incr(oddNs))) | | active(oddNs) | → | mark(incr(pairNs)) |
| active(incr(cons(X, XS))) | → | mark(cons(s(X), incr(XS))) | | active(take(0, XS)) | → | mark(nil) |
| active(take(s(N), cons(X, XS))) | → | mark(cons(X, take(N, XS))) | | active(zip(nil, XS)) | → | mark(nil) |
| active(zip(X, nil)) | → | mark(nil) | | active(zip(cons(X, XS), cons(Y, YS))) | → | mark(cons(pair(X, Y), zip(XS, YS))) |
| active(tail(cons(X, XS))) | → | mark(XS) | | active(repItems(nil)) | → | mark(nil) |
| active(repItems(cons(X, XS))) | → | mark(cons(X, cons(X, repItems(XS)))) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(incr(X)) | → | incr(active(X)) | | active(s(X)) | → | s(active(X)) |
| active(take(X1, X2)) | → | take(active(X1), X2) | | active(take(X1, X2)) | → | take(X1, active(X2)) |
| active(zip(X1, X2)) | → | zip(active(X1), X2) | | active(zip(X1, X2)) | → | zip(X1, active(X2)) |
| active(pair(X1, X2)) | → | pair(active(X1), X2) | | active(pair(X1, X2)) | → | pair(X1, active(X2)) |
| active(tail(X)) | → | tail(active(X)) | | active(repItems(X)) | → | repItems(active(X)) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | incr(mark(X)) | → | mark(incr(X)) |
| s(mark(X)) | → | mark(s(X)) | | take(mark(X1), X2) | → | mark(take(X1, X2)) |
| take(X1, mark(X2)) | → | mark(take(X1, X2)) | | zip(mark(X1), X2) | → | mark(zip(X1, X2)) |
| zip(X1, mark(X2)) | → | mark(zip(X1, X2)) | | pair(mark(X1), X2) | → | mark(pair(X1, X2)) |
| pair(X1, mark(X2)) | → | mark(pair(X1, X2)) | | tail(mark(X)) | → | mark(tail(X)) |
| repItems(mark(X)) | → | mark(repItems(X)) | | proper(pairNs) | → | ok(pairNs) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(incr(X)) | → | incr(proper(X)) | | proper(oddNs) | → | ok(oddNs) |
| proper(s(X)) | → | s(proper(X)) | | proper(take(X1, X2)) | → | take(proper(X1), proper(X2)) |
| proper(nil) | → | ok(nil) | | proper(zip(X1, X2)) | → | zip(proper(X1), proper(X2)) |
| proper(pair(X1, X2)) | → | pair(proper(X1), proper(X2)) | | proper(tail(X)) | → | tail(proper(X)) |
| proper(repItems(X)) | → | repItems(proper(X)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| incr(ok(X)) | → | ok(incr(X)) | | s(ok(X)) | → | ok(s(X)) |
| take(ok(X1), ok(X2)) | → | ok(take(X1, X2)) | | zip(ok(X1), ok(X2)) | → | ok(zip(X1, X2)) |
| pair(ok(X1), ok(X2)) | → | ok(pair(X1, X2)) | | tail(ok(X)) | → | ok(tail(X)) |
| repItems(ok(X)) | → | ok(repItems(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: zip, pair, mark, tail, 0, pairNs, s, repItems, take, active, ok, proper, incr, oddNs, nil, cons, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| active#(repItems(X)) | → | active#(X) | | active#(zip(X1, X2)) | → | active#(X2) |
| active#(incr(X)) | → | active#(X) | | active#(pair(X1, X2)) | → | active#(X2) |
| active#(s(X)) | → | active#(X) | | active#(pair(X1, X2)) | → | active#(X1) |
| active#(take(X1, X2)) | → | active#(X2) | | active#(tail(X)) | → | active#(X) |
| active#(take(X1, X2)) | → | active#(X1) | | active#(zip(X1, X2)) | → | active#(X1) |
| active#(cons(X1, X2)) | → | active#(X1) |
Problem 11: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| take#(mark(X1), X2) | → | take#(X1, X2) | | take#(X1, mark(X2)) | → | take#(X1, X2) |
| take#(ok(X1), ok(X2)) | → | take#(X1, X2) |
Rewrite Rules
| active(pairNs) | → | mark(cons(0, incr(oddNs))) | | active(oddNs) | → | mark(incr(pairNs)) |
| active(incr(cons(X, XS))) | → | mark(cons(s(X), incr(XS))) | | active(take(0, XS)) | → | mark(nil) |
| active(take(s(N), cons(X, XS))) | → | mark(cons(X, take(N, XS))) | | active(zip(nil, XS)) | → | mark(nil) |
| active(zip(X, nil)) | → | mark(nil) | | active(zip(cons(X, XS), cons(Y, YS))) | → | mark(cons(pair(X, Y), zip(XS, YS))) |
| active(tail(cons(X, XS))) | → | mark(XS) | | active(repItems(nil)) | → | mark(nil) |
| active(repItems(cons(X, XS))) | → | mark(cons(X, cons(X, repItems(XS)))) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(incr(X)) | → | incr(active(X)) | | active(s(X)) | → | s(active(X)) |
| active(take(X1, X2)) | → | take(active(X1), X2) | | active(take(X1, X2)) | → | take(X1, active(X2)) |
| active(zip(X1, X2)) | → | zip(active(X1), X2) | | active(zip(X1, X2)) | → | zip(X1, active(X2)) |
| active(pair(X1, X2)) | → | pair(active(X1), X2) | | active(pair(X1, X2)) | → | pair(X1, active(X2)) |
| active(tail(X)) | → | tail(active(X)) | | active(repItems(X)) | → | repItems(active(X)) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | incr(mark(X)) | → | mark(incr(X)) |
| s(mark(X)) | → | mark(s(X)) | | take(mark(X1), X2) | → | mark(take(X1, X2)) |
| take(X1, mark(X2)) | → | mark(take(X1, X2)) | | zip(mark(X1), X2) | → | mark(zip(X1, X2)) |
| zip(X1, mark(X2)) | → | mark(zip(X1, X2)) | | pair(mark(X1), X2) | → | mark(pair(X1, X2)) |
| pair(X1, mark(X2)) | → | mark(pair(X1, X2)) | | tail(mark(X)) | → | mark(tail(X)) |
| repItems(mark(X)) | → | mark(repItems(X)) | | proper(pairNs) | → | ok(pairNs) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(incr(X)) | → | incr(proper(X)) | | proper(oddNs) | → | ok(oddNs) |
| proper(s(X)) | → | s(proper(X)) | | proper(take(X1, X2)) | → | take(proper(X1), proper(X2)) |
| proper(nil) | → | ok(nil) | | proper(zip(X1, X2)) | → | zip(proper(X1), proper(X2)) |
| proper(pair(X1, X2)) | → | pair(proper(X1), proper(X2)) | | proper(tail(X)) | → | tail(proper(X)) |
| proper(repItems(X)) | → | repItems(proper(X)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| incr(ok(X)) | → | ok(incr(X)) | | s(ok(X)) | → | ok(s(X)) |
| take(ok(X1), ok(X2)) | → | ok(take(X1, X2)) | | zip(ok(X1), ok(X2)) | → | ok(zip(X1, X2)) |
| pair(ok(X1), ok(X2)) | → | ok(pair(X1, X2)) | | tail(ok(X)) | → | ok(tail(X)) |
| repItems(ok(X)) | → | ok(repItems(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: zip, pair, mark, tail, 0, pairNs, s, repItems, take, active, ok, proper, incr, oddNs, nil, cons, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| take#(mark(X1), X2) | → | take#(X1, X2) | | take#(ok(X1), ok(X2)) | → | take#(X1, X2) |
Problem 15: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| take#(X1, mark(X2)) | → | take#(X1, X2) |
Rewrite Rules
| active(pairNs) | → | mark(cons(0, incr(oddNs))) | | active(oddNs) | → | mark(incr(pairNs)) |
| active(incr(cons(X, XS))) | → | mark(cons(s(X), incr(XS))) | | active(take(0, XS)) | → | mark(nil) |
| active(take(s(N), cons(X, XS))) | → | mark(cons(X, take(N, XS))) | | active(zip(nil, XS)) | → | mark(nil) |
| active(zip(X, nil)) | → | mark(nil) | | active(zip(cons(X, XS), cons(Y, YS))) | → | mark(cons(pair(X, Y), zip(XS, YS))) |
| active(tail(cons(X, XS))) | → | mark(XS) | | active(repItems(nil)) | → | mark(nil) |
| active(repItems(cons(X, XS))) | → | mark(cons(X, cons(X, repItems(XS)))) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(incr(X)) | → | incr(active(X)) | | active(s(X)) | → | s(active(X)) |
| active(take(X1, X2)) | → | take(active(X1), X2) | | active(take(X1, X2)) | → | take(X1, active(X2)) |
| active(zip(X1, X2)) | → | zip(active(X1), X2) | | active(zip(X1, X2)) | → | zip(X1, active(X2)) |
| active(pair(X1, X2)) | → | pair(active(X1), X2) | | active(pair(X1, X2)) | → | pair(X1, active(X2)) |
| active(tail(X)) | → | tail(active(X)) | | active(repItems(X)) | → | repItems(active(X)) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | incr(mark(X)) | → | mark(incr(X)) |
| s(mark(X)) | → | mark(s(X)) | | take(mark(X1), X2) | → | mark(take(X1, X2)) |
| take(X1, mark(X2)) | → | mark(take(X1, X2)) | | zip(mark(X1), X2) | → | mark(zip(X1, X2)) |
| zip(X1, mark(X2)) | → | mark(zip(X1, X2)) | | pair(mark(X1), X2) | → | mark(pair(X1, X2)) |
| pair(X1, mark(X2)) | → | mark(pair(X1, X2)) | | tail(mark(X)) | → | mark(tail(X)) |
| repItems(mark(X)) | → | mark(repItems(X)) | | proper(pairNs) | → | ok(pairNs) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(incr(X)) | → | incr(proper(X)) | | proper(oddNs) | → | ok(oddNs) |
| proper(s(X)) | → | s(proper(X)) | | proper(take(X1, X2)) | → | take(proper(X1), proper(X2)) |
| proper(nil) | → | ok(nil) | | proper(zip(X1, X2)) | → | zip(proper(X1), proper(X2)) |
| proper(pair(X1, X2)) | → | pair(proper(X1), proper(X2)) | | proper(tail(X)) | → | tail(proper(X)) |
| proper(repItems(X)) | → | repItems(proper(X)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| incr(ok(X)) | → | ok(incr(X)) | | s(ok(X)) | → | ok(s(X)) |
| take(ok(X1), ok(X2)) | → | ok(take(X1, X2)) | | zip(ok(X1), ok(X2)) | → | ok(zip(X1, X2)) |
| pair(ok(X1), ok(X2)) | → | ok(pair(X1, X2)) | | tail(ok(X)) | → | ok(tail(X)) |
| repItems(ok(X)) | → | ok(repItems(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: zip, pair, mark, tail, 0, pairNs, s, repItems, take, active, ok, proper, incr, oddNs, top, cons, nil
Strategy
Polynomial Interpretation
- 0: 0
- active(x): 0
- cons(x,y): 0
- incr(x): 0
- mark(x): x + 2
- nil: 0
- oddNs: 0
- ok(x): 0
- pair(x,y): 0
- pairNs: 0
- proper(x): 0
- repItems(x): 0
- s(x): 0
- tail(x): 0
- take(x,y): 0
- take#(x,y): y + 1
- top(x): 0
- zip(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:
| take#(X1, mark(X2)) | → | take#(X1, X2) |
Problem 12: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| repItems#(ok(X)) | → | repItems#(X) | | repItems#(mark(X)) | → | repItems#(X) |
Rewrite Rules
| active(pairNs) | → | mark(cons(0, incr(oddNs))) | | active(oddNs) | → | mark(incr(pairNs)) |
| active(incr(cons(X, XS))) | → | mark(cons(s(X), incr(XS))) | | active(take(0, XS)) | → | mark(nil) |
| active(take(s(N), cons(X, XS))) | → | mark(cons(X, take(N, XS))) | | active(zip(nil, XS)) | → | mark(nil) |
| active(zip(X, nil)) | → | mark(nil) | | active(zip(cons(X, XS), cons(Y, YS))) | → | mark(cons(pair(X, Y), zip(XS, YS))) |
| active(tail(cons(X, XS))) | → | mark(XS) | | active(repItems(nil)) | → | mark(nil) |
| active(repItems(cons(X, XS))) | → | mark(cons(X, cons(X, repItems(XS)))) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(incr(X)) | → | incr(active(X)) | | active(s(X)) | → | s(active(X)) |
| active(take(X1, X2)) | → | take(active(X1), X2) | | active(take(X1, X2)) | → | take(X1, active(X2)) |
| active(zip(X1, X2)) | → | zip(active(X1), X2) | | active(zip(X1, X2)) | → | zip(X1, active(X2)) |
| active(pair(X1, X2)) | → | pair(active(X1), X2) | | active(pair(X1, X2)) | → | pair(X1, active(X2)) |
| active(tail(X)) | → | tail(active(X)) | | active(repItems(X)) | → | repItems(active(X)) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | incr(mark(X)) | → | mark(incr(X)) |
| s(mark(X)) | → | mark(s(X)) | | take(mark(X1), X2) | → | mark(take(X1, X2)) |
| take(X1, mark(X2)) | → | mark(take(X1, X2)) | | zip(mark(X1), X2) | → | mark(zip(X1, X2)) |
| zip(X1, mark(X2)) | → | mark(zip(X1, X2)) | | pair(mark(X1), X2) | → | mark(pair(X1, X2)) |
| pair(X1, mark(X2)) | → | mark(pair(X1, X2)) | | tail(mark(X)) | → | mark(tail(X)) |
| repItems(mark(X)) | → | mark(repItems(X)) | | proper(pairNs) | → | ok(pairNs) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(incr(X)) | → | incr(proper(X)) | | proper(oddNs) | → | ok(oddNs) |
| proper(s(X)) | → | s(proper(X)) | | proper(take(X1, X2)) | → | take(proper(X1), proper(X2)) |
| proper(nil) | → | ok(nil) | | proper(zip(X1, X2)) | → | zip(proper(X1), proper(X2)) |
| proper(pair(X1, X2)) | → | pair(proper(X1), proper(X2)) | | proper(tail(X)) | → | tail(proper(X)) |
| proper(repItems(X)) | → | repItems(proper(X)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| incr(ok(X)) | → | ok(incr(X)) | | s(ok(X)) | → | ok(s(X)) |
| take(ok(X1), ok(X2)) | → | ok(take(X1, X2)) | | zip(ok(X1), ok(X2)) | → | ok(zip(X1, X2)) |
| pair(ok(X1), ok(X2)) | → | ok(pair(X1, X2)) | | tail(ok(X)) | → | ok(tail(X)) |
| repItems(ok(X)) | → | ok(repItems(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: zip, pair, mark, tail, 0, pairNs, s, repItems, take, active, ok, proper, incr, oddNs, nil, cons, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| repItems#(ok(X)) | → | repItems#(X) | | repItems#(mark(X)) | → | repItems#(X) |