TIMEOUT
The TRS could not be proven terminating. The proof attempt took 60000 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (2922ms).
| – Problem 2 was processed with processor SubtermCriterion (2ms).
| – Problem 3 was processed with processor SubtermCriterion (1ms).
| – Problem 4 was processed with processor SubtermCriterion (1ms).
| – Problem 5 was processed with processor SubtermCriterion (4ms).
| – Problem 6 was processed with processor SubtermCriterion (1ms).
| – Problem 7 was processed with processor SubtermCriterion (1ms).
| – Problem 8 remains open; application of the following processors failed [SubtermCriterion (1ms), DependencyGraph (8ms), PolynomialLinearRange4iUR (3333ms), DependencyGraph (4ms), PolynomialLinearRange8NegiUR (10000ms), DependencyGraph (18ms), ReductionPairSAT (4030ms), DependencyGraph (44ms), ReductionPairSAT (3859ms), DependencyGraph (4ms), SizeChangePrinciple (timeout)].
| – Problem 9 was processed with processor SubtermCriterion (1ms).
| | – Problem 12 was processed with processor ReductionPairSAT (69ms).
| – Problem 10 was processed with processor SubtermCriterion (1ms).
| | – Problem 13 was processed with processor ReductionPairSAT (79ms).
| – Problem 11 was processed with processor SubtermCriterion (3ms).
The following open problems remain:
Open Dependency Pair Problem 8
Dependency Pairs
| top#(mark(X)) | → | top#(proper(X)) | | top#(ok(X)) | → | top#(active(X)) |
Rewrite Rules
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(head(cons(X, XS))) | → | mark(X) |
| active(2nd(cons(X, XS))) | → | mark(head(XS)) | | active(take(0, XS)) | → | mark(nil) |
| active(take(s(N), cons(X, XS))) | → | mark(cons(X, take(N, XS))) | | active(sel(0, cons(X, XS))) | → | mark(X) |
| active(sel(s(N), cons(X, XS))) | → | mark(sel(N, XS)) | | active(from(X)) | → | from(active(X)) |
| active(cons(X1, X2)) | → | cons(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(head(X)) | → | head(active(X)) | | active(2nd(X)) | → | 2nd(active(X)) |
| active(take(X1, X2)) | → | take(active(X1), X2) | | active(take(X1, X2)) | → | take(X1, active(X2)) |
| active(sel(X1, X2)) | → | sel(active(X1), X2) | | active(sel(X1, X2)) | → | sel(X1, active(X2)) |
| from(mark(X)) | → | mark(from(X)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | head(mark(X)) | → | mark(head(X)) |
| 2nd(mark(X)) | → | mark(2nd(X)) | | take(mark(X1), X2) | → | mark(take(X1, X2)) |
| take(X1, mark(X2)) | → | mark(take(X1, X2)) | | sel(mark(X1), X2) | → | mark(sel(X1, X2)) |
| sel(X1, mark(X2)) | → | mark(sel(X1, X2)) | | proper(from(X)) | → | from(proper(X)) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(s(X)) | → | s(proper(X)) |
| proper(head(X)) | → | head(proper(X)) | | proper(2nd(X)) | → | 2nd(proper(X)) |
| proper(take(X1, X2)) | → | take(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(nil) | → | ok(nil) | | proper(sel(X1, X2)) | → | sel(proper(X1), proper(X2)) |
| from(ok(X)) | → | ok(from(X)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | head(ok(X)) | → | ok(head(X)) |
| 2nd(ok(X)) | → | ok(2nd(X)) | | take(ok(X1), ok(X2)) | → | ok(take(X1, X2)) |
| sel(ok(X1), ok(X2)) | → | ok(sel(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: 2nd, mark, from, 0, s, take, active, ok, proper, head, sel, top, cons, nil
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| proper#(cons(X1, X2)) | → | proper#(X1) | | top#(ok(X)) | → | top#(active(X)) |
| active#(2nd(X)) | → | 2nd#(active(X)) | | cons#(ok(X1), ok(X2)) | → | cons#(X1, X2) |
| from#(ok(X)) | → | from#(X) | | active#(cons(X1, X2)) | → | cons#(active(X1), X2) |
| 2nd#(ok(X)) | → | 2nd#(X) | | active#(take(X1, X2)) | → | take#(active(X1), X2) |
| active#(take(s(N), cons(X, XS))) | → | cons#(X, take(N, XS)) | | active#(sel(X1, X2)) | → | active#(X2) |
| active#(head(X)) | → | active#(X) | | top#(mark(X)) | → | proper#(X) |
| proper#(from(X)) | → | proper#(X) | | active#(2nd(X)) | → | active#(X) |
| top#(mark(X)) | → | top#(proper(X)) | | proper#(cons(X1, X2)) | → | proper#(X2) |
| active#(take(X1, X2)) | → | active#(X2) | | proper#(2nd(X)) | → | 2nd#(proper(X)) |
| take#(X1, mark(X2)) | → | take#(X1, X2) | | sel#(X1, mark(X2)) | → | sel#(X1, X2) |
| active#(from(X)) | → | s#(X) | | head#(ok(X)) | → | head#(X) |
| proper#(s(X)) | → | proper#(X) | | proper#(head(X)) | → | head#(proper(X)) |
| sel#(ok(X1), ok(X2)) | → | sel#(X1, X2) | | proper#(take(X1, X2)) | → | take#(proper(X1), proper(X2)) |
| active#(cons(X1, X2)) | → | active#(X1) | | sel#(mark(X1), X2) | → | sel#(X1, X2) |
| take#(mark(X1), X2) | → | take#(X1, X2) | | proper#(head(X)) | → | proper#(X) |
| cons#(mark(X1), X2) | → | cons#(X1, X2) | | 2nd#(mark(X)) | → | 2nd#(X) |
| active#(from(X)) | → | from#(active(X)) | | from#(mark(X)) | → | from#(X) |
| top#(ok(X)) | → | active#(X) | | proper#(sel(X1, X2)) | → | sel#(proper(X1), proper(X2)) |
| proper#(2nd(X)) | → | proper#(X) | | active#(sel(X1, X2)) | → | active#(X1) |
| active#(sel(s(N), cons(X, XS))) | → | sel#(N, XS) | | proper#(take(X1, X2)) | → | proper#(X1) |
| active#(from(X)) | → | cons#(X, from(s(X))) | | proper#(from(X)) | → | from#(proper(X)) |
| proper#(sel(X1, X2)) | → | proper#(X2) | | active#(take(s(N), cons(X, XS))) | → | take#(N, XS) |
| take#(ok(X1), ok(X2)) | → | take#(X1, X2) | | active#(2nd(cons(X, XS))) | → | head#(XS) |
| active#(sel(X1, X2)) | → | sel#(X1, active(X2)) | | active#(sel(X1, X2)) | → | sel#(active(X1), X2) |
| head#(mark(X)) | → | head#(X) | | active#(from(X)) | → | active#(X) |
| proper#(take(X1, X2)) | → | proper#(X2) | | active#(head(X)) | → | head#(active(X)) |
| active#(take(X1, X2)) | → | active#(X1) | | active#(s(X)) | → | s#(active(X)) |
| s#(ok(X)) | → | s#(X) | | s#(mark(X)) | → | s#(X) |
| proper#(sel(X1, X2)) | → | proper#(X1) | | proper#(cons(X1, X2)) | → | cons#(proper(X1), proper(X2)) |
| active#(s(X)) | → | active#(X) | | active#(take(X1, X2)) | → | take#(X1, active(X2)) |
| proper#(s(X)) | → | s#(proper(X)) | | active#(from(X)) | → | from#(s(X)) |
Rewrite Rules
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(head(cons(X, XS))) | → | mark(X) |
| active(2nd(cons(X, XS))) | → | mark(head(XS)) | | active(take(0, XS)) | → | mark(nil) |
| active(take(s(N), cons(X, XS))) | → | mark(cons(X, take(N, XS))) | | active(sel(0, cons(X, XS))) | → | mark(X) |
| active(sel(s(N), cons(X, XS))) | → | mark(sel(N, XS)) | | active(from(X)) | → | from(active(X)) |
| active(cons(X1, X2)) | → | cons(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(head(X)) | → | head(active(X)) | | active(2nd(X)) | → | 2nd(active(X)) |
| active(take(X1, X2)) | → | take(active(X1), X2) | | active(take(X1, X2)) | → | take(X1, active(X2)) |
| active(sel(X1, X2)) | → | sel(active(X1), X2) | | active(sel(X1, X2)) | → | sel(X1, active(X2)) |
| from(mark(X)) | → | mark(from(X)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | head(mark(X)) | → | mark(head(X)) |
| 2nd(mark(X)) | → | mark(2nd(X)) | | take(mark(X1), X2) | → | mark(take(X1, X2)) |
| take(X1, mark(X2)) | → | mark(take(X1, X2)) | | sel(mark(X1), X2) | → | mark(sel(X1, X2)) |
| sel(X1, mark(X2)) | → | mark(sel(X1, X2)) | | proper(from(X)) | → | from(proper(X)) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(s(X)) | → | s(proper(X)) |
| proper(head(X)) | → | head(proper(X)) | | proper(2nd(X)) | → | 2nd(proper(X)) |
| proper(take(X1, X2)) | → | take(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(nil) | → | ok(nil) | | proper(sel(X1, X2)) | → | sel(proper(X1), proper(X2)) |
| from(ok(X)) | → | ok(from(X)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | head(ok(X)) | → | ok(head(X)) |
| 2nd(ok(X)) | → | ok(2nd(X)) | | take(ok(X1), ok(X2)) | → | ok(take(X1, X2)) |
| sel(ok(X1), ok(X2)) | → | ok(sel(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: 2nd, mark, from, 0, s, take, active, ok, proper, head, sel, nil, top, cons
Strategy
The following SCCs where found
| cons#(mark(X1), X2) → cons#(X1, X2) | cons#(ok(X1), ok(X2)) → cons#(X1, X2) |
| sel#(mark(X1), X2) → sel#(X1, X2) | sel#(ok(X1), ok(X2)) → sel#(X1, X2) |
| sel#(X1, mark(X2)) → sel#(X1, X2) |
| head#(ok(X)) → head#(X) | head#(mark(X)) → head#(X) |
| active#(sel(X1, X2)) → active#(X2) | active#(sel(X1, X2)) → active#(X1) |
| active#(from(X)) → active#(X) | active#(s(X)) → active#(X) |
| active#(take(X1, X2)) → active#(X2) | active#(take(X1, X2)) → active#(X1) |
| active#(head(X)) → active#(X) | active#(2nd(X)) → active#(X) |
| active#(cons(X1, X2)) → active#(X1) |
| proper#(sel(X1, X2)) → proper#(X1) | proper#(s(X)) → proper#(X) |
| proper#(head(X)) → proper#(X) | proper#(2nd(X)) → proper#(X) |
| proper#(cons(X1, X2)) → proper#(X1) | proper#(cons(X1, X2)) → proper#(X2) |
| proper#(take(X1, X2)) → proper#(X1) | proper#(take(X1, X2)) → proper#(X2) |
| proper#(sel(X1, X2)) → proper#(X2) | proper#(from(X)) → proper#(X) |
| take#(mark(X1), X2) → take#(X1, X2) | take#(X1, mark(X2)) → take#(X1, X2) |
| take#(ok(X1), ok(X2)) → take#(X1, X2) |
| from#(mark(X)) → from#(X) | from#(ok(X)) → from#(X) |
| top#(mark(X)) → top#(proper(X)) | top#(ok(X)) → top#(active(X)) |
| 2nd#(ok(X)) → 2nd#(X) | 2nd#(mark(X)) → 2nd#(X) |
| s#(mark(X)) → s#(X) | s#(ok(X)) → s#(X) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| head#(ok(X)) | → | head#(X) | | head#(mark(X)) | → | head#(X) |
Rewrite Rules
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(head(cons(X, XS))) | → | mark(X) |
| active(2nd(cons(X, XS))) | → | mark(head(XS)) | | active(take(0, XS)) | → | mark(nil) |
| active(take(s(N), cons(X, XS))) | → | mark(cons(X, take(N, XS))) | | active(sel(0, cons(X, XS))) | → | mark(X) |
| active(sel(s(N), cons(X, XS))) | → | mark(sel(N, XS)) | | active(from(X)) | → | from(active(X)) |
| active(cons(X1, X2)) | → | cons(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(head(X)) | → | head(active(X)) | | active(2nd(X)) | → | 2nd(active(X)) |
| active(take(X1, X2)) | → | take(active(X1), X2) | | active(take(X1, X2)) | → | take(X1, active(X2)) |
| active(sel(X1, X2)) | → | sel(active(X1), X2) | | active(sel(X1, X2)) | → | sel(X1, active(X2)) |
| from(mark(X)) | → | mark(from(X)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | head(mark(X)) | → | mark(head(X)) |
| 2nd(mark(X)) | → | mark(2nd(X)) | | take(mark(X1), X2) | → | mark(take(X1, X2)) |
| take(X1, mark(X2)) | → | mark(take(X1, X2)) | | sel(mark(X1), X2) | → | mark(sel(X1, X2)) |
| sel(X1, mark(X2)) | → | mark(sel(X1, X2)) | | proper(from(X)) | → | from(proper(X)) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(s(X)) | → | s(proper(X)) |
| proper(head(X)) | → | head(proper(X)) | | proper(2nd(X)) | → | 2nd(proper(X)) |
| proper(take(X1, X2)) | → | take(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(nil) | → | ok(nil) | | proper(sel(X1, X2)) | → | sel(proper(X1), proper(X2)) |
| from(ok(X)) | → | ok(from(X)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | head(ok(X)) | → | ok(head(X)) |
| 2nd(ok(X)) | → | ok(2nd(X)) | | take(ok(X1), ok(X2)) | → | ok(take(X1, X2)) |
| sel(ok(X1), ok(X2)) | → | ok(sel(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: 2nd, mark, from, 0, s, take, active, ok, proper, head, sel, nil, top, cons
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| head#(ok(X)) | → | head#(X) | | head#(mark(X)) | → | head#(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(from(X)) | → | mark(cons(X, from(s(X)))) | | active(head(cons(X, XS))) | → | mark(X) |
| active(2nd(cons(X, XS))) | → | mark(head(XS)) | | active(take(0, XS)) | → | mark(nil) |
| active(take(s(N), cons(X, XS))) | → | mark(cons(X, take(N, XS))) | | active(sel(0, cons(X, XS))) | → | mark(X) |
| active(sel(s(N), cons(X, XS))) | → | mark(sel(N, XS)) | | active(from(X)) | → | from(active(X)) |
| active(cons(X1, X2)) | → | cons(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(head(X)) | → | head(active(X)) | | active(2nd(X)) | → | 2nd(active(X)) |
| active(take(X1, X2)) | → | take(active(X1), X2) | | active(take(X1, X2)) | → | take(X1, active(X2)) |
| active(sel(X1, X2)) | → | sel(active(X1), X2) | | active(sel(X1, X2)) | → | sel(X1, active(X2)) |
| from(mark(X)) | → | mark(from(X)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | head(mark(X)) | → | mark(head(X)) |
| 2nd(mark(X)) | → | mark(2nd(X)) | | take(mark(X1), X2) | → | mark(take(X1, X2)) |
| take(X1, mark(X2)) | → | mark(take(X1, X2)) | | sel(mark(X1), X2) | → | mark(sel(X1, X2)) |
| sel(X1, mark(X2)) | → | mark(sel(X1, X2)) | | proper(from(X)) | → | from(proper(X)) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(s(X)) | → | s(proper(X)) |
| proper(head(X)) | → | head(proper(X)) | | proper(2nd(X)) | → | 2nd(proper(X)) |
| proper(take(X1, X2)) | → | take(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(nil) | → | ok(nil) | | proper(sel(X1, X2)) | → | sel(proper(X1), proper(X2)) |
| from(ok(X)) | → | ok(from(X)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | head(ok(X)) | → | ok(head(X)) |
| 2nd(ok(X)) | → | ok(2nd(X)) | | take(ok(X1), ok(X2)) | → | ok(take(X1, X2)) |
| sel(ok(X1), ok(X2)) | → | ok(sel(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: 2nd, mark, from, 0, s, take, active, ok, proper, head, sel, nil, top, cons
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
| s#(mark(X)) | → | s#(X) | | s#(ok(X)) | → | s#(X) |
Rewrite Rules
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(head(cons(X, XS))) | → | mark(X) |
| active(2nd(cons(X, XS))) | → | mark(head(XS)) | | active(take(0, XS)) | → | mark(nil) |
| active(take(s(N), cons(X, XS))) | → | mark(cons(X, take(N, XS))) | | active(sel(0, cons(X, XS))) | → | mark(X) |
| active(sel(s(N), cons(X, XS))) | → | mark(sel(N, XS)) | | active(from(X)) | → | from(active(X)) |
| active(cons(X1, X2)) | → | cons(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(head(X)) | → | head(active(X)) | | active(2nd(X)) | → | 2nd(active(X)) |
| active(take(X1, X2)) | → | take(active(X1), X2) | | active(take(X1, X2)) | → | take(X1, active(X2)) |
| active(sel(X1, X2)) | → | sel(active(X1), X2) | | active(sel(X1, X2)) | → | sel(X1, active(X2)) |
| from(mark(X)) | → | mark(from(X)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | head(mark(X)) | → | mark(head(X)) |
| 2nd(mark(X)) | → | mark(2nd(X)) | | take(mark(X1), X2) | → | mark(take(X1, X2)) |
| take(X1, mark(X2)) | → | mark(take(X1, X2)) | | sel(mark(X1), X2) | → | mark(sel(X1, X2)) |
| sel(X1, mark(X2)) | → | mark(sel(X1, X2)) | | proper(from(X)) | → | from(proper(X)) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(s(X)) | → | s(proper(X)) |
| proper(head(X)) | → | head(proper(X)) | | proper(2nd(X)) | → | 2nd(proper(X)) |
| proper(take(X1, X2)) | → | take(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(nil) | → | ok(nil) | | proper(sel(X1, X2)) | → | sel(proper(X1), proper(X2)) |
| from(ok(X)) | → | ok(from(X)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | head(ok(X)) | → | ok(head(X)) |
| 2nd(ok(X)) | → | ok(2nd(X)) | | take(ok(X1), ok(X2)) | → | ok(take(X1, X2)) |
| sel(ok(X1), ok(X2)) | → | ok(sel(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: 2nd, mark, from, 0, s, take, active, ok, proper, head, sel, nil, top, cons
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 5: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| proper#(sel(X1, X2)) | → | proper#(X1) | | proper#(s(X)) | → | proper#(X) |
| proper#(head(X)) | → | proper#(X) | | proper#(2nd(X)) | → | proper#(X) |
| proper#(cons(X1, X2)) | → | proper#(X1) | | proper#(cons(X1, X2)) | → | proper#(X2) |
| proper#(take(X1, X2)) | → | proper#(X1) | | proper#(take(X1, X2)) | → | proper#(X2) |
| proper#(sel(X1, X2)) | → | proper#(X2) | | proper#(from(X)) | → | proper#(X) |
Rewrite Rules
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(head(cons(X, XS))) | → | mark(X) |
| active(2nd(cons(X, XS))) | → | mark(head(XS)) | | active(take(0, XS)) | → | mark(nil) |
| active(take(s(N), cons(X, XS))) | → | mark(cons(X, take(N, XS))) | | active(sel(0, cons(X, XS))) | → | mark(X) |
| active(sel(s(N), cons(X, XS))) | → | mark(sel(N, XS)) | | active(from(X)) | → | from(active(X)) |
| active(cons(X1, X2)) | → | cons(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(head(X)) | → | head(active(X)) | | active(2nd(X)) | → | 2nd(active(X)) |
| active(take(X1, X2)) | → | take(active(X1), X2) | | active(take(X1, X2)) | → | take(X1, active(X2)) |
| active(sel(X1, X2)) | → | sel(active(X1), X2) | | active(sel(X1, X2)) | → | sel(X1, active(X2)) |
| from(mark(X)) | → | mark(from(X)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | head(mark(X)) | → | mark(head(X)) |
| 2nd(mark(X)) | → | mark(2nd(X)) | | take(mark(X1), X2) | → | mark(take(X1, X2)) |
| take(X1, mark(X2)) | → | mark(take(X1, X2)) | | sel(mark(X1), X2) | → | mark(sel(X1, X2)) |
| sel(X1, mark(X2)) | → | mark(sel(X1, X2)) | | proper(from(X)) | → | from(proper(X)) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(s(X)) | → | s(proper(X)) |
| proper(head(X)) | → | head(proper(X)) | | proper(2nd(X)) | → | 2nd(proper(X)) |
| proper(take(X1, X2)) | → | take(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(nil) | → | ok(nil) | | proper(sel(X1, X2)) | → | sel(proper(X1), proper(X2)) |
| from(ok(X)) | → | ok(from(X)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | head(ok(X)) | → | ok(head(X)) |
| 2nd(ok(X)) | → | ok(2nd(X)) | | take(ok(X1), ok(X2)) | → | ok(take(X1, X2)) |
| sel(ok(X1), ok(X2)) | → | ok(sel(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: 2nd, mark, from, 0, s, take, active, ok, proper, head, sel, nil, top, cons
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| proper#(sel(X1, X2)) | → | proper#(X1) | | proper#(s(X)) | → | proper#(X) |
| proper#(cons(X1, X2)) | → | proper#(X1) | | proper#(head(X)) | → | proper#(X) |
| proper#(2nd(X)) | → | proper#(X) | | proper#(cons(X1, X2)) | → | proper#(X2) |
| proper#(take(X1, X2)) | → | proper#(X1) | | proper#(take(X1, X2)) | → | proper#(X2) |
| proper#(sel(X1, X2)) | → | proper#(X2) | | proper#(from(X)) | → | proper#(X) |
Problem 6: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| from#(mark(X)) | → | from#(X) | | from#(ok(X)) | → | from#(X) |
Rewrite Rules
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(head(cons(X, XS))) | → | mark(X) |
| active(2nd(cons(X, XS))) | → | mark(head(XS)) | | active(take(0, XS)) | → | mark(nil) |
| active(take(s(N), cons(X, XS))) | → | mark(cons(X, take(N, XS))) | | active(sel(0, cons(X, XS))) | → | mark(X) |
| active(sel(s(N), cons(X, XS))) | → | mark(sel(N, XS)) | | active(from(X)) | → | from(active(X)) |
| active(cons(X1, X2)) | → | cons(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(head(X)) | → | head(active(X)) | | active(2nd(X)) | → | 2nd(active(X)) |
| active(take(X1, X2)) | → | take(active(X1), X2) | | active(take(X1, X2)) | → | take(X1, active(X2)) |
| active(sel(X1, X2)) | → | sel(active(X1), X2) | | active(sel(X1, X2)) | → | sel(X1, active(X2)) |
| from(mark(X)) | → | mark(from(X)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | head(mark(X)) | → | mark(head(X)) |
| 2nd(mark(X)) | → | mark(2nd(X)) | | take(mark(X1), X2) | → | mark(take(X1, X2)) |
| take(X1, mark(X2)) | → | mark(take(X1, X2)) | | sel(mark(X1), X2) | → | mark(sel(X1, X2)) |
| sel(X1, mark(X2)) | → | mark(sel(X1, X2)) | | proper(from(X)) | → | from(proper(X)) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(s(X)) | → | s(proper(X)) |
| proper(head(X)) | → | head(proper(X)) | | proper(2nd(X)) | → | 2nd(proper(X)) |
| proper(take(X1, X2)) | → | take(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(nil) | → | ok(nil) | | proper(sel(X1, X2)) | → | sel(proper(X1), proper(X2)) |
| from(ok(X)) | → | ok(from(X)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | head(ok(X)) | → | ok(head(X)) |
| 2nd(ok(X)) | → | ok(2nd(X)) | | take(ok(X1), ok(X2)) | → | ok(take(X1, X2)) |
| sel(ok(X1), ok(X2)) | → | ok(sel(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: 2nd, mark, from, 0, s, take, active, ok, proper, head, sel, nil, top, cons
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| from#(mark(X)) | → | from#(X) | | from#(ok(X)) | → | from#(X) |
Problem 7: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| active#(sel(X1, X2)) | → | active#(X2) | | active#(sel(X1, X2)) | → | active#(X1) |
| active#(from(X)) | → | active#(X) | | active#(s(X)) | → | active#(X) |
| active#(take(X1, X2)) | → | active#(X2) | | active#(take(X1, X2)) | → | active#(X1) |
| active#(head(X)) | → | active#(X) | | active#(2nd(X)) | → | active#(X) |
| active#(cons(X1, X2)) | → | active#(X1) |
Rewrite Rules
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(head(cons(X, XS))) | → | mark(X) |
| active(2nd(cons(X, XS))) | → | mark(head(XS)) | | active(take(0, XS)) | → | mark(nil) |
| active(take(s(N), cons(X, XS))) | → | mark(cons(X, take(N, XS))) | | active(sel(0, cons(X, XS))) | → | mark(X) |
| active(sel(s(N), cons(X, XS))) | → | mark(sel(N, XS)) | | active(from(X)) | → | from(active(X)) |
| active(cons(X1, X2)) | → | cons(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(head(X)) | → | head(active(X)) | | active(2nd(X)) | → | 2nd(active(X)) |
| active(take(X1, X2)) | → | take(active(X1), X2) | | active(take(X1, X2)) | → | take(X1, active(X2)) |
| active(sel(X1, X2)) | → | sel(active(X1), X2) | | active(sel(X1, X2)) | → | sel(X1, active(X2)) |
| from(mark(X)) | → | mark(from(X)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | head(mark(X)) | → | mark(head(X)) |
| 2nd(mark(X)) | → | mark(2nd(X)) | | take(mark(X1), X2) | → | mark(take(X1, X2)) |
| take(X1, mark(X2)) | → | mark(take(X1, X2)) | | sel(mark(X1), X2) | → | mark(sel(X1, X2)) |
| sel(X1, mark(X2)) | → | mark(sel(X1, X2)) | | proper(from(X)) | → | from(proper(X)) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(s(X)) | → | s(proper(X)) |
| proper(head(X)) | → | head(proper(X)) | | proper(2nd(X)) | → | 2nd(proper(X)) |
| proper(take(X1, X2)) | → | take(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(nil) | → | ok(nil) | | proper(sel(X1, X2)) | → | sel(proper(X1), proper(X2)) |
| from(ok(X)) | → | ok(from(X)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | head(ok(X)) | → | ok(head(X)) |
| 2nd(ok(X)) | → | ok(2nd(X)) | | take(ok(X1), ok(X2)) | → | ok(take(X1, X2)) |
| sel(ok(X1), ok(X2)) | → | ok(sel(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: 2nd, mark, from, 0, s, take, active, ok, proper, head, sel, nil, top, cons
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| active#(sel(X1, X2)) | → | active#(X2) | | active#(from(X)) | → | active#(X) |
| active#(sel(X1, X2)) | → | active#(X1) | | active#(s(X)) | → | active#(X) |
| active#(take(X1, X2)) | → | active#(X2) | | active#(take(X1, X2)) | → | active#(X1) |
| active#(head(X)) | → | active#(X) | | active#(2nd(X)) | → | active#(X) |
| active#(cons(X1, X2)) | → | active#(X1) |
Problem 9: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| sel#(mark(X1), X2) | → | sel#(X1, X2) | | sel#(ok(X1), ok(X2)) | → | sel#(X1, X2) |
| sel#(X1, mark(X2)) | → | sel#(X1, X2) |
Rewrite Rules
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(head(cons(X, XS))) | → | mark(X) |
| active(2nd(cons(X, XS))) | → | mark(head(XS)) | | active(take(0, XS)) | → | mark(nil) |
| active(take(s(N), cons(X, XS))) | → | mark(cons(X, take(N, XS))) | | active(sel(0, cons(X, XS))) | → | mark(X) |
| active(sel(s(N), cons(X, XS))) | → | mark(sel(N, XS)) | | active(from(X)) | → | from(active(X)) |
| active(cons(X1, X2)) | → | cons(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(head(X)) | → | head(active(X)) | | active(2nd(X)) | → | 2nd(active(X)) |
| active(take(X1, X2)) | → | take(active(X1), X2) | | active(take(X1, X2)) | → | take(X1, active(X2)) |
| active(sel(X1, X2)) | → | sel(active(X1), X2) | | active(sel(X1, X2)) | → | sel(X1, active(X2)) |
| from(mark(X)) | → | mark(from(X)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | head(mark(X)) | → | mark(head(X)) |
| 2nd(mark(X)) | → | mark(2nd(X)) | | take(mark(X1), X2) | → | mark(take(X1, X2)) |
| take(X1, mark(X2)) | → | mark(take(X1, X2)) | | sel(mark(X1), X2) | → | mark(sel(X1, X2)) |
| sel(X1, mark(X2)) | → | mark(sel(X1, X2)) | | proper(from(X)) | → | from(proper(X)) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(s(X)) | → | s(proper(X)) |
| proper(head(X)) | → | head(proper(X)) | | proper(2nd(X)) | → | 2nd(proper(X)) |
| proper(take(X1, X2)) | → | take(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(nil) | → | ok(nil) | | proper(sel(X1, X2)) | → | sel(proper(X1), proper(X2)) |
| from(ok(X)) | → | ok(from(X)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | head(ok(X)) | → | ok(head(X)) |
| 2nd(ok(X)) | → | ok(2nd(X)) | | take(ok(X1), ok(X2)) | → | ok(take(X1, X2)) |
| sel(ok(X1), ok(X2)) | → | ok(sel(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: 2nd, mark, from, 0, s, take, active, ok, proper, head, sel, nil, top, cons
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| sel#(mark(X1), X2) | → | sel#(X1, X2) | | sel#(ok(X1), ok(X2)) | → | sel#(X1, X2) |
Problem 12: ReductionPairSAT
Dependency Pair Problem
Dependency Pairs
| sel#(X1, mark(X2)) | → | sel#(X1, X2) |
Rewrite Rules
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(head(cons(X, XS))) | → | mark(X) |
| active(2nd(cons(X, XS))) | → | mark(head(XS)) | | active(take(0, XS)) | → | mark(nil) |
| active(take(s(N), cons(X, XS))) | → | mark(cons(X, take(N, XS))) | | active(sel(0, cons(X, XS))) | → | mark(X) |
| active(sel(s(N), cons(X, XS))) | → | mark(sel(N, XS)) | | active(from(X)) | → | from(active(X)) |
| active(cons(X1, X2)) | → | cons(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(head(X)) | → | head(active(X)) | | active(2nd(X)) | → | 2nd(active(X)) |
| active(take(X1, X2)) | → | take(active(X1), X2) | | active(take(X1, X2)) | → | take(X1, active(X2)) |
| active(sel(X1, X2)) | → | sel(active(X1), X2) | | active(sel(X1, X2)) | → | sel(X1, active(X2)) |
| from(mark(X)) | → | mark(from(X)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | head(mark(X)) | → | mark(head(X)) |
| 2nd(mark(X)) | → | mark(2nd(X)) | | take(mark(X1), X2) | → | mark(take(X1, X2)) |
| take(X1, mark(X2)) | → | mark(take(X1, X2)) | | sel(mark(X1), X2) | → | mark(sel(X1, X2)) |
| sel(X1, mark(X2)) | → | mark(sel(X1, X2)) | | proper(from(X)) | → | from(proper(X)) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(s(X)) | → | s(proper(X)) |
| proper(head(X)) | → | head(proper(X)) | | proper(2nd(X)) | → | 2nd(proper(X)) |
| proper(take(X1, X2)) | → | take(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(nil) | → | ok(nil) | | proper(sel(X1, X2)) | → | sel(proper(X1), proper(X2)) |
| from(ok(X)) | → | ok(from(X)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | head(ok(X)) | → | ok(head(X)) |
| 2nd(ok(X)) | → | ok(2nd(X)) | | take(ok(X1), ok(X2)) | → | ok(take(X1, X2)) |
| sel(ok(X1), ok(X2)) | → | ok(sel(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: 2nd, mark, from, 0, s, take, active, ok, proper, head, sel, cons, nil, top
Strategy
Function Precedence
mark < 2nd = sel# = from = 0 = s = take = active = ok = proper = head = sel = top = cons = nil
Argument Filtering
2nd: collapses to 1
mark: 1
sel#: collapses to 2
from: collapses to 1
0: all arguments are removed from 0
s: collapses to 1
take: collapses to 2
active: collapses to 1
ok: collapses to 1
proper: collapses to 1
head: 1
sel: 1 2
top: all arguments are removed from top
cons: all arguments are removed from cons
nil: all arguments are removed from nil
Status
mark: lexicographic with permutation 1 → 1
0: multiset
head: lexicographic with permutation 1 → 1
sel: lexicographic with permutation 1 → 1 2 → 2
top: multiset
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.
| sel#(X1, mark(X2)) → sel#(X1, X2) |
Problem 10: 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(from(X)) | → | mark(cons(X, from(s(X)))) | | active(head(cons(X, XS))) | → | mark(X) |
| active(2nd(cons(X, XS))) | → | mark(head(XS)) | | active(take(0, XS)) | → | mark(nil) |
| active(take(s(N), cons(X, XS))) | → | mark(cons(X, take(N, XS))) | | active(sel(0, cons(X, XS))) | → | mark(X) |
| active(sel(s(N), cons(X, XS))) | → | mark(sel(N, XS)) | | active(from(X)) | → | from(active(X)) |
| active(cons(X1, X2)) | → | cons(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(head(X)) | → | head(active(X)) | | active(2nd(X)) | → | 2nd(active(X)) |
| active(take(X1, X2)) | → | take(active(X1), X2) | | active(take(X1, X2)) | → | take(X1, active(X2)) |
| active(sel(X1, X2)) | → | sel(active(X1), X2) | | active(sel(X1, X2)) | → | sel(X1, active(X2)) |
| from(mark(X)) | → | mark(from(X)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | head(mark(X)) | → | mark(head(X)) |
| 2nd(mark(X)) | → | mark(2nd(X)) | | take(mark(X1), X2) | → | mark(take(X1, X2)) |
| take(X1, mark(X2)) | → | mark(take(X1, X2)) | | sel(mark(X1), X2) | → | mark(sel(X1, X2)) |
| sel(X1, mark(X2)) | → | mark(sel(X1, X2)) | | proper(from(X)) | → | from(proper(X)) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(s(X)) | → | s(proper(X)) |
| proper(head(X)) | → | head(proper(X)) | | proper(2nd(X)) | → | 2nd(proper(X)) |
| proper(take(X1, X2)) | → | take(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(nil) | → | ok(nil) | | proper(sel(X1, X2)) | → | sel(proper(X1), proper(X2)) |
| from(ok(X)) | → | ok(from(X)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | head(ok(X)) | → | ok(head(X)) |
| 2nd(ok(X)) | → | ok(2nd(X)) | | take(ok(X1), ok(X2)) | → | ok(take(X1, X2)) |
| sel(ok(X1), ok(X2)) | → | ok(sel(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: 2nd, mark, from, 0, s, take, active, ok, proper, head, sel, nil, top, cons
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 13: ReductionPairSAT
Dependency Pair Problem
Dependency Pairs
| take#(X1, mark(X2)) | → | take#(X1, X2) |
Rewrite Rules
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(head(cons(X, XS))) | → | mark(X) |
| active(2nd(cons(X, XS))) | → | mark(head(XS)) | | active(take(0, XS)) | → | mark(nil) |
| active(take(s(N), cons(X, XS))) | → | mark(cons(X, take(N, XS))) | | active(sel(0, cons(X, XS))) | → | mark(X) |
| active(sel(s(N), cons(X, XS))) | → | mark(sel(N, XS)) | | active(from(X)) | → | from(active(X)) |
| active(cons(X1, X2)) | → | cons(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(head(X)) | → | head(active(X)) | | active(2nd(X)) | → | 2nd(active(X)) |
| active(take(X1, X2)) | → | take(active(X1), X2) | | active(take(X1, X2)) | → | take(X1, active(X2)) |
| active(sel(X1, X2)) | → | sel(active(X1), X2) | | active(sel(X1, X2)) | → | sel(X1, active(X2)) |
| from(mark(X)) | → | mark(from(X)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | head(mark(X)) | → | mark(head(X)) |
| 2nd(mark(X)) | → | mark(2nd(X)) | | take(mark(X1), X2) | → | mark(take(X1, X2)) |
| take(X1, mark(X2)) | → | mark(take(X1, X2)) | | sel(mark(X1), X2) | → | mark(sel(X1, X2)) |
| sel(X1, mark(X2)) | → | mark(sel(X1, X2)) | | proper(from(X)) | → | from(proper(X)) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(s(X)) | → | s(proper(X)) |
| proper(head(X)) | → | head(proper(X)) | | proper(2nd(X)) | → | 2nd(proper(X)) |
| proper(take(X1, X2)) | → | take(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(nil) | → | ok(nil) | | proper(sel(X1, X2)) | → | sel(proper(X1), proper(X2)) |
| from(ok(X)) | → | ok(from(X)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | head(ok(X)) | → | ok(head(X)) |
| 2nd(ok(X)) | → | ok(2nd(X)) | | take(ok(X1), ok(X2)) | → | ok(take(X1, X2)) |
| sel(ok(X1), ok(X2)) | → | ok(sel(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: 2nd, mark, from, 0, s, take, active, ok, proper, head, sel, cons, nil, top
Strategy
Function Precedence
2nd = mark = from = take# = 0 = s = take = active = ok = proper = head = sel = top = cons = nil
Argument Filtering
2nd: all arguments are removed from 2nd
mark: 1
from: all arguments are removed from from
take#: 2
0: all arguments are removed from 0
s: all arguments are removed from s
take: 1
active: all arguments are removed from active
ok: all arguments are removed from ok
proper: all arguments are removed from proper
head: 1
sel: all arguments are removed from sel
top: all arguments are removed from top
cons: all arguments are removed from cons
nil: all arguments are removed from nil
Status
2nd: multiset
mark: multiset
from: multiset
take#: lexicographic with permutation 2 → 1
0: multiset
s: multiset
take: lexicographic with permutation 1 → 1
active: multiset
ok: multiset
proper: multiset
head: lexicographic with permutation 1 → 1
sel: multiset
top: multiset
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.
| take#(X1, mark(X2)) → take#(X1, X2) |
Problem 11: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| 2nd#(ok(X)) | → | 2nd#(X) | | 2nd#(mark(X)) | → | 2nd#(X) |
Rewrite Rules
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(head(cons(X, XS))) | → | mark(X) |
| active(2nd(cons(X, XS))) | → | mark(head(XS)) | | active(take(0, XS)) | → | mark(nil) |
| active(take(s(N), cons(X, XS))) | → | mark(cons(X, take(N, XS))) | | active(sel(0, cons(X, XS))) | → | mark(X) |
| active(sel(s(N), cons(X, XS))) | → | mark(sel(N, XS)) | | active(from(X)) | → | from(active(X)) |
| active(cons(X1, X2)) | → | cons(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(head(X)) | → | head(active(X)) | | active(2nd(X)) | → | 2nd(active(X)) |
| active(take(X1, X2)) | → | take(active(X1), X2) | | active(take(X1, X2)) | → | take(X1, active(X2)) |
| active(sel(X1, X2)) | → | sel(active(X1), X2) | | active(sel(X1, X2)) | → | sel(X1, active(X2)) |
| from(mark(X)) | → | mark(from(X)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | head(mark(X)) | → | mark(head(X)) |
| 2nd(mark(X)) | → | mark(2nd(X)) | | take(mark(X1), X2) | → | mark(take(X1, X2)) |
| take(X1, mark(X2)) | → | mark(take(X1, X2)) | | sel(mark(X1), X2) | → | mark(sel(X1, X2)) |
| sel(X1, mark(X2)) | → | mark(sel(X1, X2)) | | proper(from(X)) | → | from(proper(X)) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(s(X)) | → | s(proper(X)) |
| proper(head(X)) | → | head(proper(X)) | | proper(2nd(X)) | → | 2nd(proper(X)) |
| proper(take(X1, X2)) | → | take(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| proper(nil) | → | ok(nil) | | proper(sel(X1, X2)) | → | sel(proper(X1), proper(X2)) |
| from(ok(X)) | → | ok(from(X)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | head(ok(X)) | → | ok(head(X)) |
| 2nd(ok(X)) | → | ok(2nd(X)) | | take(ok(X1), ok(X2)) | → | ok(take(X1, X2)) |
| sel(ok(X1), ok(X2)) | → | ok(sel(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: 2nd, mark, from, 0, s, take, active, ok, proper, head, sel, nil, top, cons
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| 2nd#(ok(X)) | → | 2nd#(X) | | 2nd#(mark(X)) | → | 2nd#(X) |