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 (1517ms).
| – Problem 2 was processed with processor SubtermCriterion (2ms).
| – Problem 3 was processed with processor SubtermCriterion (6ms).
| – Problem 4 was processed with processor SubtermCriterion (1ms).
| – Problem 5 was processed with processor SubtermCriterion (0ms).
| | – Problem 9 was processed with processor ReductionPairSAT (78ms).
| | | – Problem 10 was processed with processor ReductionPairSAT (69ms).
| – 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 (0ms), DependencyGraph (356ms), PolynomialLinearRange4iUR (5000ms), DependencyGraph (280ms), PolynomialLinearRange8NegiUR (15000ms), DependencyGraph (270ms), ReductionPairSAT (6583ms), DependencyGraph (276ms), ReductionPairSAT (6657ms), DependencyGraph (271ms), ReductionPairSAT (6365ms), DependencyGraph (281ms), SizeChangePrinciple (timeout)].
The following open problems remain:
Open Dependency Pair Problem 8
Dependency Pairs
| mark#(cons(X1, X2)) | → | active#(cons(mark(X1), X2)) | | mark#(head(X)) | → | active#(head(mark(X))) |
| active#(tail(cons(X, L))) | → | mark#(L) | | mark#(tail(X)) | → | active#(tail(mark(X))) |
| mark#(nil) | → | active#(nil) | | mark#(head(X)) | → | mark#(X) |
| active#(nats) | → | mark#(adx(zeros)) | | mark#(s(X)) | → | mark#(X) |
| mark#(zeros) | → | active#(zeros) | | active#(incr(nil)) | → | mark#(nil) |
| active#(adx(nil)) | → | mark#(nil) | | mark#(0) | → | active#(0) |
| mark#(s(X)) | → | active#(s(mark(X))) | | active#(incr(cons(X, L))) | → | mark#(cons(s(X), incr(L))) |
| mark#(nats) | → | active#(nats) | | mark#(cons(X1, X2)) | → | mark#(X1) |
| mark#(incr(X)) | → | active#(incr(mark(X))) | | mark#(incr(X)) | → | mark#(X) |
| active#(adx(cons(X, L))) | → | mark#(incr(cons(X, adx(L)))) | | active#(head(cons(X, L))) | → | mark#(X) |
| mark#(adx(X)) | → | mark#(X) | | mark#(tail(X)) | → | mark#(X) |
| active#(zeros) | → | mark#(cons(0, zeros)) | | mark#(adx(X)) | → | active#(adx(mark(X))) |
Rewrite Rules
| active(incr(nil)) | → | mark(nil) | | active(incr(cons(X, L))) | → | mark(cons(s(X), incr(L))) |
| active(adx(nil)) | → | mark(nil) | | active(adx(cons(X, L))) | → | mark(incr(cons(X, adx(L)))) |
| active(nats) | → | mark(adx(zeros)) | | active(zeros) | → | mark(cons(0, zeros)) |
| active(head(cons(X, L))) | → | mark(X) | | active(tail(cons(X, L))) | → | mark(L) |
| mark(incr(X)) | → | active(incr(mark(X))) | | mark(nil) | → | active(nil) |
| mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(adx(X)) | → | active(adx(mark(X))) | | mark(nats) | → | active(nats) |
| mark(zeros) | → | active(zeros) | | mark(0) | → | active(0) |
| mark(head(X)) | → | active(head(mark(X))) | | mark(tail(X)) | → | active(tail(mark(X))) |
| incr(mark(X)) | → | incr(X) | | incr(active(X)) | → | incr(X) |
| cons(mark(X1), X2) | → | cons(X1, X2) | | cons(X1, mark(X2)) | → | cons(X1, X2) |
| cons(active(X1), X2) | → | cons(X1, X2) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| adx(mark(X)) | → | adx(X) | | adx(active(X)) | → | adx(X) |
| head(mark(X)) | → | head(X) | | head(active(X)) | → | head(X) |
| tail(mark(X)) | → | tail(X) | | tail(active(X)) | → | tail(X) |
Original Signature
Termination of terms over the following signature is verified: nats, 0, s, zeros, adx, active, mark, incr, head, tail, cons, nil
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| mark#(cons(X1, X2)) | → | active#(cons(mark(X1), X2)) | | active#(incr(cons(X, L))) | → | incr#(L) |
| mark#(s(X)) | → | s#(mark(X)) | | mark#(tail(X)) | → | active#(tail(mark(X))) |
| active#(incr(cons(X, L))) | → | cons#(s(X), incr(L)) | | active#(nats) | → | mark#(adx(zeros)) |
| mark#(s(X)) | → | mark#(X) | | adx#(active(X)) | → | adx#(X) |
| mark#(cons(X1, X2)) | → | mark#(X1) | | tail#(mark(X)) | → | tail#(X) |
| mark#(incr(X)) | → | active#(incr(mark(X))) | | cons#(X1, active(X2)) | → | cons#(X1, X2) |
| mark#(adx(X)) | → | mark#(X) | | mark#(tail(X)) | → | mark#(X) |
| head#(active(X)) | → | head#(X) | | active#(adx(cons(X, L))) | → | incr#(cons(X, adx(L))) |
| mark#(adx(X)) | → | active#(adx(mark(X))) | | mark#(head(X)) | → | active#(head(mark(X))) |
| incr#(active(X)) | → | incr#(X) | | cons#(mark(X1), X2) | → | cons#(X1, X2) |
| mark#(tail(X)) | → | tail#(mark(X)) | | active#(tail(cons(X, L))) | → | mark#(L) |
| mark#(head(X)) | → | mark#(X) | | mark#(nil) | → | active#(nil) |
| active#(incr(cons(X, L))) | → | s#(X) | | tail#(active(X)) | → | tail#(X) |
| incr#(mark(X)) | → | incr#(X) | | mark#(incr(X)) | → | incr#(mark(X)) |
| cons#(X1, mark(X2)) | → | cons#(X1, X2) | | mark#(adx(X)) | → | adx#(mark(X)) |
| mark#(zeros) | → | active#(zeros) | | mark#(cons(X1, X2)) | → | cons#(mark(X1), X2) |
| active#(incr(nil)) | → | mark#(nil) | | active#(adx(nil)) | → | mark#(nil) |
| mark#(0) | → | active#(0) | | mark#(s(X)) | → | active#(s(mark(X))) |
| active#(incr(cons(X, L))) | → | mark#(cons(s(X), incr(L))) | | mark#(nats) | → | active#(nats) |
| head#(mark(X)) | → | head#(X) | | active#(adx(cons(X, L))) | → | cons#(X, adx(L)) |
| active#(adx(cons(X, L))) | → | adx#(L) | | cons#(active(X1), X2) | → | cons#(X1, X2) |
| mark#(incr(X)) | → | mark#(X) | | mark#(head(X)) | → | head#(mark(X)) |
| active#(nats) | → | adx#(zeros) | | s#(mark(X)) | → | s#(X) |
| adx#(mark(X)) | → | adx#(X) | | active#(adx(cons(X, L))) | → | mark#(incr(cons(X, adx(L)))) |
| active#(head(cons(X, L))) | → | mark#(X) | | active#(zeros) | → | cons#(0, zeros) |
| active#(zeros) | → | mark#(cons(0, zeros)) | | s#(active(X)) | → | s#(X) |
Rewrite Rules
| active(incr(nil)) | → | mark(nil) | | active(incr(cons(X, L))) | → | mark(cons(s(X), incr(L))) |
| active(adx(nil)) | → | mark(nil) | | active(adx(cons(X, L))) | → | mark(incr(cons(X, adx(L)))) |
| active(nats) | → | mark(adx(zeros)) | | active(zeros) | → | mark(cons(0, zeros)) |
| active(head(cons(X, L))) | → | mark(X) | | active(tail(cons(X, L))) | → | mark(L) |
| mark(incr(X)) | → | active(incr(mark(X))) | | mark(nil) | → | active(nil) |
| mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(adx(X)) | → | active(adx(mark(X))) | | mark(nats) | → | active(nats) |
| mark(zeros) | → | active(zeros) | | mark(0) | → | active(0) |
| mark(head(X)) | → | active(head(mark(X))) | | mark(tail(X)) | → | active(tail(mark(X))) |
| incr(mark(X)) | → | incr(X) | | incr(active(X)) | → | incr(X) |
| cons(mark(X1), X2) | → | cons(X1, X2) | | cons(X1, mark(X2)) | → | cons(X1, X2) |
| cons(active(X1), X2) | → | cons(X1, X2) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| adx(mark(X)) | → | adx(X) | | adx(active(X)) | → | adx(X) |
| head(mark(X)) | → | head(X) | | head(active(X)) | → | head(X) |
| tail(mark(X)) | → | tail(X) | | tail(active(X)) | → | tail(X) |
Original Signature
Termination of terms over the following signature is verified: nats, 0, s, zeros, active, adx, mark, incr, head, tail, nil, cons
Strategy
The following SCCs where found
| active#(adx(nil)) → mark#(nil) | mark#(head(X)) → active#(head(mark(X))) |
| mark#(cons(X1, X2)) → active#(cons(mark(X1), X2)) | mark#(0) → active#(0) |
| mark#(s(X)) → active#(s(mark(X))) | mark#(nats) → active#(nats) |
| active#(incr(cons(X, L))) → mark#(cons(s(X), incr(L))) | mark#(cons(X1, X2)) → mark#(X1) |
| mark#(incr(X)) → active#(incr(mark(X))) | mark#(incr(X)) → mark#(X) |
| active#(tail(cons(X, L))) → mark#(L) | mark#(tail(X)) → active#(tail(mark(X))) |
| mark#(head(X)) → mark#(X) | mark#(nil) → active#(nil) |
| active#(adx(cons(X, L))) → mark#(incr(cons(X, adx(L)))) | mark#(adx(X)) → mark#(X) |
| active#(head(cons(X, L))) → mark#(X) | mark#(tail(X)) → mark#(X) |
| active#(nats) → mark#(adx(zeros)) | active#(zeros) → mark#(cons(0, zeros)) |
| mark#(s(X)) → mark#(X) | mark#(zeros) → active#(zeros) |
| mark#(adx(X)) → active#(adx(mark(X))) | active#(incr(nil)) → mark#(nil) |
| adx#(mark(X)) → adx#(X) | adx#(active(X)) → adx#(X) |
| tail#(active(X)) → tail#(X) | tail#(mark(X)) → tail#(X) |
| s#(mark(X)) → s#(X) | s#(active(X)) → s#(X) |
| cons#(X1, active(X2)) → cons#(X1, X2) | cons#(mark(X1), X2) → cons#(X1, X2) |
| cons#(X1, mark(X2)) → cons#(X1, X2) | cons#(active(X1), X2) → cons#(X1, X2) |
| incr#(active(X)) → incr#(X) | incr#(mark(X)) → incr#(X) |
| head#(mark(X)) → head#(X) | head#(active(X)) → head#(X) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| s#(mark(X)) | → | s#(X) | | s#(active(X)) | → | s#(X) |
Rewrite Rules
| active(incr(nil)) | → | mark(nil) | | active(incr(cons(X, L))) | → | mark(cons(s(X), incr(L))) |
| active(adx(nil)) | → | mark(nil) | | active(adx(cons(X, L))) | → | mark(incr(cons(X, adx(L)))) |
| active(nats) | → | mark(adx(zeros)) | | active(zeros) | → | mark(cons(0, zeros)) |
| active(head(cons(X, L))) | → | mark(X) | | active(tail(cons(X, L))) | → | mark(L) |
| mark(incr(X)) | → | active(incr(mark(X))) | | mark(nil) | → | active(nil) |
| mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(adx(X)) | → | active(adx(mark(X))) | | mark(nats) | → | active(nats) |
| mark(zeros) | → | active(zeros) | | mark(0) | → | active(0) |
| mark(head(X)) | → | active(head(mark(X))) | | mark(tail(X)) | → | active(tail(mark(X))) |
| incr(mark(X)) | → | incr(X) | | incr(active(X)) | → | incr(X) |
| cons(mark(X1), X2) | → | cons(X1, X2) | | cons(X1, mark(X2)) | → | cons(X1, X2) |
| cons(active(X1), X2) | → | cons(X1, X2) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| adx(mark(X)) | → | adx(X) | | adx(active(X)) | → | adx(X) |
| head(mark(X)) | → | head(X) | | head(active(X)) | → | head(X) |
| tail(mark(X)) | → | tail(X) | | tail(active(X)) | → | tail(X) |
Original Signature
Termination of terms over the following signature is verified: nats, 0, s, zeros, active, adx, mark, incr, head, tail, nil, cons
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| s#(mark(X)) | → | s#(X) | | s#(active(X)) | → | s#(X) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| head#(mark(X)) | → | head#(X) | | head#(active(X)) | → | head#(X) |
Rewrite Rules
| active(incr(nil)) | → | mark(nil) | | active(incr(cons(X, L))) | → | mark(cons(s(X), incr(L))) |
| active(adx(nil)) | → | mark(nil) | | active(adx(cons(X, L))) | → | mark(incr(cons(X, adx(L)))) |
| active(nats) | → | mark(adx(zeros)) | | active(zeros) | → | mark(cons(0, zeros)) |
| active(head(cons(X, L))) | → | mark(X) | | active(tail(cons(X, L))) | → | mark(L) |
| mark(incr(X)) | → | active(incr(mark(X))) | | mark(nil) | → | active(nil) |
| mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(adx(X)) | → | active(adx(mark(X))) | | mark(nats) | → | active(nats) |
| mark(zeros) | → | active(zeros) | | mark(0) | → | active(0) |
| mark(head(X)) | → | active(head(mark(X))) | | mark(tail(X)) | → | active(tail(mark(X))) |
| incr(mark(X)) | → | incr(X) | | incr(active(X)) | → | incr(X) |
| cons(mark(X1), X2) | → | cons(X1, X2) | | cons(X1, mark(X2)) | → | cons(X1, X2) |
| cons(active(X1), X2) | → | cons(X1, X2) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| adx(mark(X)) | → | adx(X) | | adx(active(X)) | → | adx(X) |
| head(mark(X)) | → | head(X) | | head(active(X)) | → | head(X) |
| tail(mark(X)) | → | tail(X) | | tail(active(X)) | → | tail(X) |
Original Signature
Termination of terms over the following signature is verified: nats, 0, s, zeros, active, adx, mark, incr, head, tail, nil, cons
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| head#(mark(X)) | → | head#(X) | | head#(active(X)) | → | head#(X) |
Problem 4: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| incr#(active(X)) | → | incr#(X) | | incr#(mark(X)) | → | incr#(X) |
Rewrite Rules
| active(incr(nil)) | → | mark(nil) | | active(incr(cons(X, L))) | → | mark(cons(s(X), incr(L))) |
| active(adx(nil)) | → | mark(nil) | | active(adx(cons(X, L))) | → | mark(incr(cons(X, adx(L)))) |
| active(nats) | → | mark(adx(zeros)) | | active(zeros) | → | mark(cons(0, zeros)) |
| active(head(cons(X, L))) | → | mark(X) | | active(tail(cons(X, L))) | → | mark(L) |
| mark(incr(X)) | → | active(incr(mark(X))) | | mark(nil) | → | active(nil) |
| mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(adx(X)) | → | active(adx(mark(X))) | | mark(nats) | → | active(nats) |
| mark(zeros) | → | active(zeros) | | mark(0) | → | active(0) |
| mark(head(X)) | → | active(head(mark(X))) | | mark(tail(X)) | → | active(tail(mark(X))) |
| incr(mark(X)) | → | incr(X) | | incr(active(X)) | → | incr(X) |
| cons(mark(X1), X2) | → | cons(X1, X2) | | cons(X1, mark(X2)) | → | cons(X1, X2) |
| cons(active(X1), X2) | → | cons(X1, X2) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| adx(mark(X)) | → | adx(X) | | adx(active(X)) | → | adx(X) |
| head(mark(X)) | → | head(X) | | head(active(X)) | → | head(X) |
| tail(mark(X)) | → | tail(X) | | tail(active(X)) | → | tail(X) |
Original Signature
Termination of terms over the following signature is verified: nats, 0, s, zeros, active, adx, mark, incr, head, tail, nil, cons
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| incr#(active(X)) | → | incr#(X) | | incr#(mark(X)) | → | incr#(X) |
Problem 5: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| cons#(X1, active(X2)) | → | cons#(X1, X2) | | cons#(mark(X1), X2) | → | cons#(X1, X2) |
| cons#(X1, mark(X2)) | → | cons#(X1, X2) | | cons#(active(X1), X2) | → | cons#(X1, X2) |
Rewrite Rules
| active(incr(nil)) | → | mark(nil) | | active(incr(cons(X, L))) | → | mark(cons(s(X), incr(L))) |
| active(adx(nil)) | → | mark(nil) | | active(adx(cons(X, L))) | → | mark(incr(cons(X, adx(L)))) |
| active(nats) | → | mark(adx(zeros)) | | active(zeros) | → | mark(cons(0, zeros)) |
| active(head(cons(X, L))) | → | mark(X) | | active(tail(cons(X, L))) | → | mark(L) |
| mark(incr(X)) | → | active(incr(mark(X))) | | mark(nil) | → | active(nil) |
| mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(adx(X)) | → | active(adx(mark(X))) | | mark(nats) | → | active(nats) |
| mark(zeros) | → | active(zeros) | | mark(0) | → | active(0) |
| mark(head(X)) | → | active(head(mark(X))) | | mark(tail(X)) | → | active(tail(mark(X))) |
| incr(mark(X)) | → | incr(X) | | incr(active(X)) | → | incr(X) |
| cons(mark(X1), X2) | → | cons(X1, X2) | | cons(X1, mark(X2)) | → | cons(X1, X2) |
| cons(active(X1), X2) | → | cons(X1, X2) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| adx(mark(X)) | → | adx(X) | | adx(active(X)) | → | adx(X) |
| head(mark(X)) | → | head(X) | | head(active(X)) | → | head(X) |
| tail(mark(X)) | → | tail(X) | | tail(active(X)) | → | tail(X) |
Original Signature
Termination of terms over the following signature is verified: nats, 0, s, zeros, active, adx, mark, incr, head, tail, nil, cons
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| cons#(mark(X1), X2) | → | cons#(X1, X2) | | cons#(active(X1), X2) | → | cons#(X1, X2) |
Problem 9: ReductionPairSAT
Dependency Pair Problem
Dependency Pairs
| cons#(X1, active(X2)) | → | cons#(X1, X2) | | cons#(X1, mark(X2)) | → | cons#(X1, X2) |
Rewrite Rules
| active(incr(nil)) | → | mark(nil) | | active(incr(cons(X, L))) | → | mark(cons(s(X), incr(L))) |
| active(adx(nil)) | → | mark(nil) | | active(adx(cons(X, L))) | → | mark(incr(cons(X, adx(L)))) |
| active(nats) | → | mark(adx(zeros)) | | active(zeros) | → | mark(cons(0, zeros)) |
| active(head(cons(X, L))) | → | mark(X) | | active(tail(cons(X, L))) | → | mark(L) |
| mark(incr(X)) | → | active(incr(mark(X))) | | mark(nil) | → | active(nil) |
| mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(adx(X)) | → | active(adx(mark(X))) | | mark(nats) | → | active(nats) |
| mark(zeros) | → | active(zeros) | | mark(0) | → | active(0) |
| mark(head(X)) | → | active(head(mark(X))) | | mark(tail(X)) | → | active(tail(mark(X))) |
| incr(mark(X)) | → | incr(X) | | incr(active(X)) | → | incr(X) |
| cons(mark(X1), X2) | → | cons(X1, X2) | | cons(X1, mark(X2)) | → | cons(X1, X2) |
| cons(active(X1), X2) | → | cons(X1, X2) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| adx(mark(X)) | → | adx(X) | | adx(active(X)) | → | adx(X) |
| head(mark(X)) | → | head(X) | | head(active(X)) | → | head(X) |
| tail(mark(X)) | → | tail(X) | | tail(active(X)) | → | tail(X) |
Original Signature
Termination of terms over the following signature is verified: nats, 0, s, zeros, adx, active, mark, incr, head, tail, cons, nil
Strategy
Function Precedence
cons# < active < mark < tail = nats = 0 = s = zeros = adx = incr = head = cons = nil
Argument Filtering
mark: collapses to 1
tail: 1
nats: all arguments are removed from nats
cons#: 1 2
0: all arguments are removed from 0
s: collapses to 1
zeros: all arguments are removed from zeros
adx: all arguments are removed from adx
active: 1
incr: 1
head: collapses to 1
cons: 1 2
nil: all arguments are removed from nil
Status
tail: lexicographic with permutation 1 → 1
nats: multiset
cons#: lexicographic with permutation 1 → 1 2 → 2
0: multiset
zeros: multiset
adx: multiset
active: multiset
incr: lexicographic with permutation 1 → 1
cons: lexicographic with permutation 1 → 1 2 → 2
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.
| cons#(X1, active(X2)) → cons#(X1, X2) |
Problem 10: ReductionPairSAT
Dependency Pair Problem
Dependency Pairs
| cons#(X1, mark(X2)) | → | cons#(X1, X2) |
Rewrite Rules
| active(incr(nil)) | → | mark(nil) | | active(incr(cons(X, L))) | → | mark(cons(s(X), incr(L))) |
| active(adx(nil)) | → | mark(nil) | | active(adx(cons(X, L))) | → | mark(incr(cons(X, adx(L)))) |
| active(nats) | → | mark(adx(zeros)) | | active(zeros) | → | mark(cons(0, zeros)) |
| active(head(cons(X, L))) | → | mark(X) | | active(tail(cons(X, L))) | → | mark(L) |
| mark(incr(X)) | → | active(incr(mark(X))) | | mark(nil) | → | active(nil) |
| mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(adx(X)) | → | active(adx(mark(X))) | | mark(nats) | → | active(nats) |
| mark(zeros) | → | active(zeros) | | mark(0) | → | active(0) |
| mark(head(X)) | → | active(head(mark(X))) | | mark(tail(X)) | → | active(tail(mark(X))) |
| incr(mark(X)) | → | incr(X) | | incr(active(X)) | → | incr(X) |
| cons(mark(X1), X2) | → | cons(X1, X2) | | cons(X1, mark(X2)) | → | cons(X1, X2) |
| cons(active(X1), X2) | → | cons(X1, X2) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| adx(mark(X)) | → | adx(X) | | adx(active(X)) | → | adx(X) |
| head(mark(X)) | → | head(X) | | head(active(X)) | → | head(X) |
| tail(mark(X)) | → | tail(X) | | tail(active(X)) | → | tail(X) |
Original Signature
Termination of terms over the following signature is verified: nats, 0, s, zeros, active, adx, mark, incr, head, tail, nil, cons
Strategy
Function Precedence
mark < tail = nats = cons# = 0 = s = zeros = adx = active = incr = head = cons = nil
Argument Filtering
mark: 1
tail: collapses to 1
nats: all arguments are removed from nats
cons#: collapses to 2
0: all arguments are removed from 0
s: collapses to 1
zeros: all arguments are removed from zeros
adx: collapses to 1
active: all arguments are removed from active
incr: collapses to 1
head: collapses to 1
cons: 2
nil: all arguments are removed from nil
Status
mark: lexicographic with permutation 1 → 1
nats: multiset
0: multiset
zeros: multiset
active: multiset
cons: lexicographic with permutation 2 → 1
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.
| cons#(X1, mark(X2)) → cons#(X1, X2) |
Problem 6: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| adx#(mark(X)) | → | adx#(X) | | adx#(active(X)) | → | adx#(X) |
Rewrite Rules
| active(incr(nil)) | → | mark(nil) | | active(incr(cons(X, L))) | → | mark(cons(s(X), incr(L))) |
| active(adx(nil)) | → | mark(nil) | | active(adx(cons(X, L))) | → | mark(incr(cons(X, adx(L)))) |
| active(nats) | → | mark(adx(zeros)) | | active(zeros) | → | mark(cons(0, zeros)) |
| active(head(cons(X, L))) | → | mark(X) | | active(tail(cons(X, L))) | → | mark(L) |
| mark(incr(X)) | → | active(incr(mark(X))) | | mark(nil) | → | active(nil) |
| mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(adx(X)) | → | active(adx(mark(X))) | | mark(nats) | → | active(nats) |
| mark(zeros) | → | active(zeros) | | mark(0) | → | active(0) |
| mark(head(X)) | → | active(head(mark(X))) | | mark(tail(X)) | → | active(tail(mark(X))) |
| incr(mark(X)) | → | incr(X) | | incr(active(X)) | → | incr(X) |
| cons(mark(X1), X2) | → | cons(X1, X2) | | cons(X1, mark(X2)) | → | cons(X1, X2) |
| cons(active(X1), X2) | → | cons(X1, X2) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| adx(mark(X)) | → | adx(X) | | adx(active(X)) | → | adx(X) |
| head(mark(X)) | → | head(X) | | head(active(X)) | → | head(X) |
| tail(mark(X)) | → | tail(X) | | tail(active(X)) | → | tail(X) |
Original Signature
Termination of terms over the following signature is verified: nats, 0, s, zeros, active, adx, mark, incr, head, tail, nil, cons
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| adx#(mark(X)) | → | adx#(X) | | adx#(active(X)) | → | adx#(X) |
Problem 7: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| tail#(active(X)) | → | tail#(X) | | tail#(mark(X)) | → | tail#(X) |
Rewrite Rules
| active(incr(nil)) | → | mark(nil) | | active(incr(cons(X, L))) | → | mark(cons(s(X), incr(L))) |
| active(adx(nil)) | → | mark(nil) | | active(adx(cons(X, L))) | → | mark(incr(cons(X, adx(L)))) |
| active(nats) | → | mark(adx(zeros)) | | active(zeros) | → | mark(cons(0, zeros)) |
| active(head(cons(X, L))) | → | mark(X) | | active(tail(cons(X, L))) | → | mark(L) |
| mark(incr(X)) | → | active(incr(mark(X))) | | mark(nil) | → | active(nil) |
| mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(adx(X)) | → | active(adx(mark(X))) | | mark(nats) | → | active(nats) |
| mark(zeros) | → | active(zeros) | | mark(0) | → | active(0) |
| mark(head(X)) | → | active(head(mark(X))) | | mark(tail(X)) | → | active(tail(mark(X))) |
| incr(mark(X)) | → | incr(X) | | incr(active(X)) | → | incr(X) |
| cons(mark(X1), X2) | → | cons(X1, X2) | | cons(X1, mark(X2)) | → | cons(X1, X2) |
| cons(active(X1), X2) | → | cons(X1, X2) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| adx(mark(X)) | → | adx(X) | | adx(active(X)) | → | adx(X) |
| head(mark(X)) | → | head(X) | | head(active(X)) | → | head(X) |
| tail(mark(X)) | → | tail(X) | | tail(active(X)) | → | tail(X) |
Original Signature
Termination of terms over the following signature is verified: nats, 0, s, zeros, active, adx, mark, incr, head, tail, nil, cons
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| tail#(active(X)) | → | tail#(X) | | tail#(mark(X)) | → | tail#(X) |