TIMEOUT
The TRS could not be proven terminating. The proof attempt took 60018 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (1464ms).
| – Problem 2 remains open; application of the following processors failed [SubtermCriterion (2ms), DependencyGraph (3ms), PolynomialLinearRange4iUR (3334ms), DependencyGraph (3ms), PolynomialLinearRange8NegiUR (10000ms), DependencyGraph (4ms), ReductionPairSAT (3813ms), DependencyGraph (4ms), ReductionPairSAT (3669ms), DependencyGraph (2ms), SizeChangePrinciple (timeout)].
| – Problem 3 was processed with processor SubtermCriterion (2ms).
| | – Problem 10 was processed with processor ReductionPairSAT (50ms).
| – Problem 4 was processed with processor SubtermCriterion (1ms).
| | – Problem 11 was processed with processor ReductionPairSAT (75ms).
| – Problem 5 was processed with processor SubtermCriterion (1ms).
| – Problem 6 was processed with processor SubtermCriterion (1ms).
| – Problem 7 was processed with processor SubtermCriterion (2ms).
| – Problem 8 was processed with processor SubtermCriterion (0ms).
| – Problem 9 was processed with processor SubtermCriterion (0ms).
The following open problems remain:
Open Dependency Pair Problem 2
Dependency Pairs
| top#(mark(X)) | → | top#(proper(X)) | | top#(ok(X)) | → | top#(active(X)) |
Rewrite Rules
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(first(0, Z)) | → | mark(nil) |
| active(first(s(X), cons(Y, Z))) | → | mark(cons(Y, first(X, Z))) | | active(sel(0, cons(X, Z))) | → | mark(X) |
| active(sel(s(X), cons(Y, Z))) | → | mark(sel(X, Z)) | | active(from(X)) | → | from(active(X)) |
| active(cons(X1, X2)) | → | cons(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(first(X1, X2)) | → | first(active(X1), X2) | | active(first(X1, X2)) | → | first(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)) | | first(mark(X1), X2) | → | mark(first(X1, X2)) |
| first(X1, mark(X2)) | → | mark(first(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(first(X1, X2)) | → | first(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)) | | first(ok(X1), ok(X2)) | → | ok(first(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: 0, s, active, ok, mark, proper, from, sel, first, top, nil, cons
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| proper#(cons(X1, X2)) | → | proper#(X1) | | top#(ok(X)) | → | top#(active(X)) |
| cons#(ok(X1), ok(X2)) | → | cons#(X1, X2) | | from#(ok(X)) | → | from#(X) |
| active#(first(s(X), cons(Y, Z))) | → | cons#(Y, first(X, Z)) | | active#(cons(X1, X2)) | → | cons#(active(X1), X2) |
| active#(sel(X1, X2)) | → | active#(X2) | | first#(mark(X1), X2) | → | first#(X1, X2) |
| top#(mark(X)) | → | proper#(X) | | proper#(from(X)) | → | proper#(X) |
| active#(first(X1, X2)) | → | active#(X2) | | active#(sel(s(X), cons(Y, Z))) | → | sel#(X, Z) |
| top#(mark(X)) | → | top#(proper(X)) | | proper#(cons(X1, X2)) | → | proper#(X2) |
| proper#(first(X1, X2)) | → | first#(proper(X1), proper(X2)) | | sel#(X1, mark(X2)) | → | sel#(X1, X2) |
| active#(first(X1, X2)) | → | active#(X1) | | proper#(first(X1, X2)) | → | proper#(X2) |
| active#(from(X)) | → | s#(X) | | proper#(s(X)) | → | proper#(X) |
| sel#(ok(X1), ok(X2)) | → | sel#(X1, X2) | | active#(cons(X1, X2)) | → | active#(X1) |
| sel#(mark(X1), X2) | → | sel#(X1, X2) | | cons#(mark(X1), X2) | → | cons#(X1, X2) |
| active#(from(X)) | → | from#(active(X)) | | active#(first(X1, X2)) | → | first#(X1, active(X2)) |
| from#(mark(X)) | → | from#(X) | | top#(ok(X)) | → | active#(X) |
| proper#(sel(X1, X2)) | → | sel#(proper(X1), proper(X2)) | | active#(first(s(X), cons(Y, Z))) | → | first#(X, Z) |
| active#(sel(X1, X2)) | → | active#(X1) | | active#(from(X)) | → | cons#(X, from(s(X))) |
| proper#(from(X)) | → | from#(proper(X)) | | proper#(sel(X1, X2)) | → | proper#(X2) |
| first#(X1, mark(X2)) | → | first#(X1, X2) | | active#(sel(X1, X2)) | → | sel#(X1, active(X2)) |
| active#(sel(X1, X2)) | → | sel#(active(X1), X2) | | active#(from(X)) | → | active#(X) |
| active#(s(X)) | → | s#(active(X)) | | s#(ok(X)) | → | s#(X) |
| s#(mark(X)) | → | s#(X) | | proper#(sel(X1, X2)) | → | proper#(X1) |
| first#(ok(X1), ok(X2)) | → | first#(X1, X2) | | active#(first(X1, X2)) | → | first#(active(X1), X2) |
| proper#(cons(X1, X2)) | → | cons#(proper(X1), proper(X2)) | | active#(s(X)) | → | active#(X) |
| proper#(s(X)) | → | s#(proper(X)) | | proper#(first(X1, X2)) | → | proper#(X1) |
| active#(from(X)) | → | from#(s(X)) |
Rewrite Rules
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(first(0, Z)) | → | mark(nil) |
| active(first(s(X), cons(Y, Z))) | → | mark(cons(Y, first(X, Z))) | | active(sel(0, cons(X, Z))) | → | mark(X) |
| active(sel(s(X), cons(Y, Z))) | → | mark(sel(X, Z)) | | active(from(X)) | → | from(active(X)) |
| active(cons(X1, X2)) | → | cons(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(first(X1, X2)) | → | first(active(X1), X2) | | active(first(X1, X2)) | → | first(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)) | | first(mark(X1), X2) | → | mark(first(X1, X2)) |
| first(X1, mark(X2)) | → | mark(first(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(first(X1, X2)) | → | first(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)) | | first(ok(X1), ok(X2)) | → | ok(first(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: 0, s, active, mark, ok, from, proper, first, sel, cons, nil, top
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) |
| active#(first(X1, X2)) → active#(X2) | active#(sel(X1, X2)) → active#(X2) |
| active#(s(X)) → active#(X) | active#(from(X)) → active#(X) |
| active#(sel(X1, X2)) → active#(X1) | active#(first(X1, X2)) → active#(X1) |
| active#(cons(X1, X2)) → active#(X1) |
| proper#(sel(X1, X2)) → proper#(X1) | proper#(first(X1, X2)) → proper#(X2) |
| proper#(s(X)) → proper#(X) | proper#(cons(X1, X2)) → proper#(X1) |
| proper#(cons(X1, X2)) → proper#(X2) | proper#(first(X1, X2)) → proper#(X1) |
| proper#(sel(X1, X2)) → proper#(X2) | proper#(from(X)) → proper#(X) |
| from#(mark(X)) → from#(X) | from#(ok(X)) → from#(X) |
| top#(mark(X)) → top#(proper(X)) | top#(ok(X)) → top#(active(X)) |
| s#(mark(X)) → s#(X) | s#(ok(X)) → s#(X) |
| first#(ok(X1), ok(X2)) → first#(X1, X2) | first#(mark(X1), X2) → first#(X1, X2) |
| first#(X1, mark(X2)) → first#(X1, X2) |
Problem 3: 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(first(0, Z)) | → | mark(nil) |
| active(first(s(X), cons(Y, Z))) | → | mark(cons(Y, first(X, Z))) | | active(sel(0, cons(X, Z))) | → | mark(X) |
| active(sel(s(X), cons(Y, Z))) | → | mark(sel(X, Z)) | | active(from(X)) | → | from(active(X)) |
| active(cons(X1, X2)) | → | cons(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(first(X1, X2)) | → | first(active(X1), X2) | | active(first(X1, X2)) | → | first(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)) | | first(mark(X1), X2) | → | mark(first(X1, X2)) |
| first(X1, mark(X2)) | → | mark(first(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(first(X1, X2)) | → | first(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)) | | first(ok(X1), ok(X2)) | → | ok(first(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: 0, s, active, mark, ok, from, proper, first, sel, cons, nil, top
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 10: 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(first(0, Z)) | → | mark(nil) |
| active(first(s(X), cons(Y, Z))) | → | mark(cons(Y, first(X, Z))) | | active(sel(0, cons(X, Z))) | → | mark(X) |
| active(sel(s(X), cons(Y, Z))) | → | mark(sel(X, Z)) | | active(from(X)) | → | from(active(X)) |
| active(cons(X1, X2)) | → | cons(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(first(X1, X2)) | → | first(active(X1), X2) | | active(first(X1, X2)) | → | first(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)) | | first(mark(X1), X2) | → | mark(first(X1, X2)) |
| first(X1, mark(X2)) | → | mark(first(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(first(X1, X2)) | → | first(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)) | | first(ok(X1), ok(X2)) | → | ok(first(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: 0, s, active, ok, mark, proper, from, sel, first, top, nil, cons
Strategy
Function Precedence
mark < sel# = from = 0 = s = active = ok = proper = first = sel = cons = nil = top
Argument Filtering
mark: 1
sel#: collapses to 2
from: collapses to 1
0: all arguments are removed from 0
s: all arguments are removed from s
active: all arguments are removed from active
ok: all arguments are removed from ok
proper: collapses to 1
first: 1 2
sel: all arguments are removed from sel
cons: 1 2
nil: all arguments are removed from nil
top: collapses to 1
Status
mark: multiset
0: multiset
s: multiset
active: multiset
ok: multiset
first: lexicographic with permutation 1 → 1 2 → 2
sel: multiset
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.
| sel#(X1, mark(X2)) → sel#(X1, X2) |
Problem 4: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| first#(ok(X1), ok(X2)) | → | first#(X1, X2) | | first#(mark(X1), X2) | → | first#(X1, X2) |
| first#(X1, mark(X2)) | → | first#(X1, X2) |
Rewrite Rules
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(first(0, Z)) | → | mark(nil) |
| active(first(s(X), cons(Y, Z))) | → | mark(cons(Y, first(X, Z))) | | active(sel(0, cons(X, Z))) | → | mark(X) |
| active(sel(s(X), cons(Y, Z))) | → | mark(sel(X, Z)) | | active(from(X)) | → | from(active(X)) |
| active(cons(X1, X2)) | → | cons(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(first(X1, X2)) | → | first(active(X1), X2) | | active(first(X1, X2)) | → | first(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)) | | first(mark(X1), X2) | → | mark(first(X1, X2)) |
| first(X1, mark(X2)) | → | mark(first(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(first(X1, X2)) | → | first(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)) | | first(ok(X1), ok(X2)) | → | ok(first(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: 0, s, active, mark, ok, from, proper, first, sel, cons, nil, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| first#(ok(X1), ok(X2)) | → | first#(X1, X2) | | first#(mark(X1), X2) | → | first#(X1, X2) |
Problem 11: ReductionPairSAT
Dependency Pair Problem
Dependency Pairs
| first#(X1, mark(X2)) | → | first#(X1, X2) |
Rewrite Rules
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(first(0, Z)) | → | mark(nil) |
| active(first(s(X), cons(Y, Z))) | → | mark(cons(Y, first(X, Z))) | | active(sel(0, cons(X, Z))) | → | mark(X) |
| active(sel(s(X), cons(Y, Z))) | → | mark(sel(X, Z)) | | active(from(X)) | → | from(active(X)) |
| active(cons(X1, X2)) | → | cons(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(first(X1, X2)) | → | first(active(X1), X2) | | active(first(X1, X2)) | → | first(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)) | | first(mark(X1), X2) | → | mark(first(X1, X2)) |
| first(X1, mark(X2)) | → | mark(first(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(first(X1, X2)) | → | first(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)) | | first(ok(X1), ok(X2)) | → | ok(first(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: 0, s, active, ok, mark, proper, from, sel, first, top, nil, cons
Strategy
Function Precedence
mark = from = first# = 0 = s = active = ok = proper = first = sel = cons = nil = top
Argument Filtering
mark: 1
from: collapses to 1
first#: 2
0: all arguments are removed from 0
s: all arguments are removed from s
active: collapses to 1
ok: all arguments are removed from ok
proper: collapses to 1
first: collapses to 1
sel: collapses to 1
cons: all arguments are removed from cons
nil: all arguments are removed from nil
top: all arguments are removed from top
Status
mark: multiset
first#: lexicographic with permutation 2 → 1
0: multiset
s: multiset
ok: multiset
cons: multiset
nil: multiset
top: 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.
| first#(X1, mark(X2)) → first#(X1, X2) |
Problem 5: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| active#(first(X1, X2)) | → | active#(X2) | | active#(sel(X1, X2)) | → | active#(X2) |
| active#(s(X)) | → | active#(X) | | active#(from(X)) | → | active#(X) |
| active#(sel(X1, X2)) | → | active#(X1) | | active#(first(X1, X2)) | → | active#(X1) |
| active#(cons(X1, X2)) | → | active#(X1) |
Rewrite Rules
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(first(0, Z)) | → | mark(nil) |
| active(first(s(X), cons(Y, Z))) | → | mark(cons(Y, first(X, Z))) | | active(sel(0, cons(X, Z))) | → | mark(X) |
| active(sel(s(X), cons(Y, Z))) | → | mark(sel(X, Z)) | | active(from(X)) | → | from(active(X)) |
| active(cons(X1, X2)) | → | cons(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(first(X1, X2)) | → | first(active(X1), X2) | | active(first(X1, X2)) | → | first(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)) | | first(mark(X1), X2) | → | mark(first(X1, X2)) |
| first(X1, mark(X2)) | → | mark(first(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(first(X1, X2)) | → | first(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)) | | first(ok(X1), ok(X2)) | → | ok(first(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: 0, s, active, mark, ok, from, proper, first, sel, cons, nil, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| active#(first(X1, X2)) | → | active#(X2) | | active#(sel(X1, X2)) | → | active#(X2) |
| active#(sel(X1, X2)) | → | active#(X1) | | active#(s(X)) | → | active#(X) |
| active#(from(X)) | → | active#(X) | | active#(first(X1, X2)) | → | active#(X1) |
| active#(cons(X1, X2)) | → | active#(X1) |
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(first(0, Z)) | → | mark(nil) |
| active(first(s(X), cons(Y, Z))) | → | mark(cons(Y, first(X, Z))) | | active(sel(0, cons(X, Z))) | → | mark(X) |
| active(sel(s(X), cons(Y, Z))) | → | mark(sel(X, Z)) | | active(from(X)) | → | from(active(X)) |
| active(cons(X1, X2)) | → | cons(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(first(X1, X2)) | → | first(active(X1), X2) | | active(first(X1, X2)) | → | first(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)) | | first(mark(X1), X2) | → | mark(first(X1, X2)) |
| first(X1, mark(X2)) | → | mark(first(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(first(X1, X2)) | → | first(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)) | | first(ok(X1), ok(X2)) | → | ok(first(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: 0, s, active, mark, ok, from, proper, first, sel, cons, nil, top
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
| proper#(sel(X1, X2)) | → | proper#(X1) | | proper#(first(X1, X2)) | → | proper#(X2) |
| proper#(s(X)) | → | proper#(X) | | proper#(cons(X1, X2)) | → | proper#(X1) |
| proper#(cons(X1, X2)) | → | proper#(X2) | | proper#(first(X1, X2)) | → | proper#(X1) |
| proper#(sel(X1, X2)) | → | proper#(X2) | | proper#(from(X)) | → | proper#(X) |
Rewrite Rules
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(first(0, Z)) | → | mark(nil) |
| active(first(s(X), cons(Y, Z))) | → | mark(cons(Y, first(X, Z))) | | active(sel(0, cons(X, Z))) | → | mark(X) |
| active(sel(s(X), cons(Y, Z))) | → | mark(sel(X, Z)) | | active(from(X)) | → | from(active(X)) |
| active(cons(X1, X2)) | → | cons(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(first(X1, X2)) | → | first(active(X1), X2) | | active(first(X1, X2)) | → | first(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)) | | first(mark(X1), X2) | → | mark(first(X1, X2)) |
| first(X1, mark(X2)) | → | mark(first(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(first(X1, X2)) | → | first(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)) | | first(ok(X1), ok(X2)) | → | ok(first(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: 0, s, active, mark, ok, from, proper, first, sel, cons, nil, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| proper#(first(X1, X2)) | → | proper#(X2) | | proper#(sel(X1, X2)) | → | proper#(X1) |
| proper#(s(X)) | → | proper#(X) | | proper#(cons(X1, X2)) | → | proper#(X1) |
| proper#(cons(X1, X2)) | → | proper#(X2) | | proper#(first(X1, X2)) | → | proper#(X1) |
| proper#(sel(X1, X2)) | → | proper#(X2) | | proper#(from(X)) | → | proper#(X) |
Problem 8: 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(first(0, Z)) | → | mark(nil) |
| active(first(s(X), cons(Y, Z))) | → | mark(cons(Y, first(X, Z))) | | active(sel(0, cons(X, Z))) | → | mark(X) |
| active(sel(s(X), cons(Y, Z))) | → | mark(sel(X, Z)) | | active(from(X)) | → | from(active(X)) |
| active(cons(X1, X2)) | → | cons(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(first(X1, X2)) | → | first(active(X1), X2) | | active(first(X1, X2)) | → | first(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)) | | first(mark(X1), X2) | → | mark(first(X1, X2)) |
| first(X1, mark(X2)) | → | mark(first(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(first(X1, X2)) | → | first(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)) | | first(ok(X1), ok(X2)) | → | ok(first(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: 0, s, active, mark, ok, from, proper, first, sel, cons, nil, 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 9: 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(first(0, Z)) | → | mark(nil) |
| active(first(s(X), cons(Y, Z))) | → | mark(cons(Y, first(X, Z))) | | active(sel(0, cons(X, Z))) | → | mark(X) |
| active(sel(s(X), cons(Y, Z))) | → | mark(sel(X, Z)) | | active(from(X)) | → | from(active(X)) |
| active(cons(X1, X2)) | → | cons(active(X1), X2) | | active(s(X)) | → | s(active(X)) |
| active(first(X1, X2)) | → | first(active(X1), X2) | | active(first(X1, X2)) | → | first(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)) | | first(mark(X1), X2) | → | mark(first(X1, X2)) |
| first(X1, mark(X2)) | → | mark(first(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(first(X1, X2)) | → | first(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)) | | first(ok(X1), ok(X2)) | → | ok(first(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: 0, s, active, mark, ok, from, proper, first, sel, cons, nil, 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) |