YES
The TRS could be proven terminating. The proof took 1353 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (227ms).
| – Problem 2 was processed with processor SubtermCriterion (8ms).
| – Problem 3 was processed with processor SubtermCriterion (23ms).
| – Problem 4 was processed with processor SubtermCriterion (5ms).
| – Problem 5 was processed with processor SubtermCriterion (1ms).
| – Problem 6 was processed with processor SubtermCriterion (0ms).
| – Problem 7 was processed with processor SubtermCriterion (1ms).
| – Problem 8 was processed with processor SubtermCriterion (0ms).
| – Problem 9 was processed with processor PolynomialLinearRange4iUR (711ms).
| | – Problem 10 was processed with processor PolynomialLinearRange4iUR (341ms).
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| h#(mark(X)) | → | h#(X) | | active#(h(X)) | → | active#(X) |
| active#(f(f(X))) | → | c#(f(g(f(X)))) | | h#(ok(X)) | → | h#(X) |
| top#(ok(X)) | → | top#(active(X)) | | g#(ok(X)) | → | g#(X) |
| d#(ok(X)) | → | d#(X) | | proper#(c(X)) | → | proper#(X) |
| top#(ok(X)) | → | active#(X) | | c#(ok(X)) | → | c#(X) |
| proper#(d(X)) | → | d#(proper(X)) | | active#(h(X)) | → | c#(d(X)) |
| active#(h(X)) | → | d#(X) | | active#(f(f(X))) | → | g#(f(X)) |
| f#(mark(X)) | → | f#(X) | | top#(mark(X)) | → | proper#(X) |
| f#(ok(X)) | → | f#(X) | | proper#(f(X)) | → | f#(proper(X)) |
| top#(mark(X)) | → | top#(proper(X)) | | proper#(f(X)) | → | proper#(X) |
| active#(c(X)) | → | d#(X) | | active#(f(f(X))) | → | f#(X) |
| active#(f(X)) | → | f#(active(X)) | | active#(f(f(X))) | → | f#(g(f(X))) |
| proper#(h(X)) | → | proper#(X) | | proper#(h(X)) | → | h#(proper(X)) |
| proper#(g(X)) | → | g#(proper(X)) | | proper#(g(X)) | → | proper#(X) |
| proper#(c(X)) | → | c#(proper(X)) | | active#(f(X)) | → | active#(X) |
| proper#(d(X)) | → | proper#(X) | | active#(h(X)) | → | h#(active(X)) |
Rewrite Rules
| active(f(f(X))) | → | mark(c(f(g(f(X))))) | | active(c(X)) | → | mark(d(X)) |
| active(h(X)) | → | mark(c(d(X))) | | active(f(X)) | → | f(active(X)) |
| active(h(X)) | → | h(active(X)) | | f(mark(X)) | → | mark(f(X)) |
| h(mark(X)) | → | mark(h(X)) | | proper(f(X)) | → | f(proper(X)) |
| proper(c(X)) | → | c(proper(X)) | | proper(g(X)) | → | g(proper(X)) |
| proper(d(X)) | → | d(proper(X)) | | proper(h(X)) | → | h(proper(X)) |
| f(ok(X)) | → | ok(f(X)) | | c(ok(X)) | → | ok(c(X)) |
| g(ok(X)) | → | ok(g(X)) | | d(ok(X)) | → | ok(d(X)) |
| h(ok(X)) | → | ok(h(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: f, g, d, c, active, mark, ok, proper, h, top
Strategy
The following SCCs where found
| f#(mark(X)) → f#(X) | f#(ok(X)) → f#(X) |
| active#(h(X)) → active#(X) | active#(f(X)) → active#(X) |
| proper#(f(X)) → proper#(X) | proper#(h(X)) → proper#(X) |
| proper#(g(X)) → proper#(X) | proper#(c(X)) → proper#(X) |
| proper#(d(X)) → proper#(X) |
| top#(mark(X)) → top#(proper(X)) | top#(ok(X)) → top#(active(X)) |
| h#(mark(X)) → h#(X) | h#(ok(X)) → h#(X) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| active#(h(X)) | → | active#(X) | | active#(f(X)) | → | active#(X) |
Rewrite Rules
| active(f(f(X))) | → | mark(c(f(g(f(X))))) | | active(c(X)) | → | mark(d(X)) |
| active(h(X)) | → | mark(c(d(X))) | | active(f(X)) | → | f(active(X)) |
| active(h(X)) | → | h(active(X)) | | f(mark(X)) | → | mark(f(X)) |
| h(mark(X)) | → | mark(h(X)) | | proper(f(X)) | → | f(proper(X)) |
| proper(c(X)) | → | c(proper(X)) | | proper(g(X)) | → | g(proper(X)) |
| proper(d(X)) | → | d(proper(X)) | | proper(h(X)) | → | h(proper(X)) |
| f(ok(X)) | → | ok(f(X)) | | c(ok(X)) | → | ok(c(X)) |
| g(ok(X)) | → | ok(g(X)) | | d(ok(X)) | → | ok(d(X)) |
| h(ok(X)) | → | ok(h(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: f, g, d, c, active, mark, ok, proper, h, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| active#(h(X)) | → | active#(X) | | active#(f(X)) | → | active#(X) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
Rewrite Rules
| active(f(f(X))) | → | mark(c(f(g(f(X))))) | | active(c(X)) | → | mark(d(X)) |
| active(h(X)) | → | mark(c(d(X))) | | active(f(X)) | → | f(active(X)) |
| active(h(X)) | → | h(active(X)) | | f(mark(X)) | → | mark(f(X)) |
| h(mark(X)) | → | mark(h(X)) | | proper(f(X)) | → | f(proper(X)) |
| proper(c(X)) | → | c(proper(X)) | | proper(g(X)) | → | g(proper(X)) |
| proper(d(X)) | → | d(proper(X)) | | proper(h(X)) | → | h(proper(X)) |
| f(ok(X)) | → | ok(f(X)) | | c(ok(X)) | → | ok(c(X)) |
| g(ok(X)) | → | ok(g(X)) | | d(ok(X)) | → | ok(d(X)) |
| h(ok(X)) | → | ok(h(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: f, g, d, c, active, mark, ok, proper, h, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
Problem 4: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
Rewrite Rules
| active(f(f(X))) | → | mark(c(f(g(f(X))))) | | active(c(X)) | → | mark(d(X)) |
| active(h(X)) | → | mark(c(d(X))) | | active(f(X)) | → | f(active(X)) |
| active(h(X)) | → | h(active(X)) | | f(mark(X)) | → | mark(f(X)) |
| h(mark(X)) | → | mark(h(X)) | | proper(f(X)) | → | f(proper(X)) |
| proper(c(X)) | → | c(proper(X)) | | proper(g(X)) | → | g(proper(X)) |
| proper(d(X)) | → | d(proper(X)) | | proper(h(X)) | → | h(proper(X)) |
| f(ok(X)) | → | ok(f(X)) | | c(ok(X)) | → | ok(c(X)) |
| g(ok(X)) | → | ok(g(X)) | | d(ok(X)) | → | ok(d(X)) |
| h(ok(X)) | → | ok(h(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: f, g, d, c, active, mark, ok, proper, h, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
Problem 5: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| h#(mark(X)) | → | h#(X) | | h#(ok(X)) | → | h#(X) |
Rewrite Rules
| active(f(f(X))) | → | mark(c(f(g(f(X))))) | | active(c(X)) | → | mark(d(X)) |
| active(h(X)) | → | mark(c(d(X))) | | active(f(X)) | → | f(active(X)) |
| active(h(X)) | → | h(active(X)) | | f(mark(X)) | → | mark(f(X)) |
| h(mark(X)) | → | mark(h(X)) | | proper(f(X)) | → | f(proper(X)) |
| proper(c(X)) | → | c(proper(X)) | | proper(g(X)) | → | g(proper(X)) |
| proper(d(X)) | → | d(proper(X)) | | proper(h(X)) | → | h(proper(X)) |
| f(ok(X)) | → | ok(f(X)) | | c(ok(X)) | → | ok(c(X)) |
| g(ok(X)) | → | ok(g(X)) | | d(ok(X)) | → | ok(d(X)) |
| h(ok(X)) | → | ok(h(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: f, g, d, c, active, mark, ok, proper, h, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| h#(mark(X)) | → | h#(X) | | h#(ok(X)) | → | h#(X) |
Problem 6: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| proper#(f(X)) | → | proper#(X) | | proper#(h(X)) | → | proper#(X) |
| proper#(g(X)) | → | proper#(X) | | proper#(c(X)) | → | proper#(X) |
| proper#(d(X)) | → | proper#(X) |
Rewrite Rules
| active(f(f(X))) | → | mark(c(f(g(f(X))))) | | active(c(X)) | → | mark(d(X)) |
| active(h(X)) | → | mark(c(d(X))) | | active(f(X)) | → | f(active(X)) |
| active(h(X)) | → | h(active(X)) | | f(mark(X)) | → | mark(f(X)) |
| h(mark(X)) | → | mark(h(X)) | | proper(f(X)) | → | f(proper(X)) |
| proper(c(X)) | → | c(proper(X)) | | proper(g(X)) | → | g(proper(X)) |
| proper(d(X)) | → | d(proper(X)) | | proper(h(X)) | → | h(proper(X)) |
| f(ok(X)) | → | ok(f(X)) | | c(ok(X)) | → | ok(c(X)) |
| g(ok(X)) | → | ok(g(X)) | | d(ok(X)) | → | ok(d(X)) |
| h(ok(X)) | → | ok(h(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: f, g, d, c, active, mark, ok, proper, h, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| proper#(f(X)) | → | proper#(X) | | proper#(h(X)) | → | proper#(X) |
| proper#(c(X)) | → | proper#(X) | | proper#(g(X)) | → | proper#(X) |
| proper#(d(X)) | → | proper#(X) |
Problem 7: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| f#(mark(X)) | → | f#(X) | | f#(ok(X)) | → | f#(X) |
Rewrite Rules
| active(f(f(X))) | → | mark(c(f(g(f(X))))) | | active(c(X)) | → | mark(d(X)) |
| active(h(X)) | → | mark(c(d(X))) | | active(f(X)) | → | f(active(X)) |
| active(h(X)) | → | h(active(X)) | | f(mark(X)) | → | mark(f(X)) |
| h(mark(X)) | → | mark(h(X)) | | proper(f(X)) | → | f(proper(X)) |
| proper(c(X)) | → | c(proper(X)) | | proper(g(X)) | → | g(proper(X)) |
| proper(d(X)) | → | d(proper(X)) | | proper(h(X)) | → | h(proper(X)) |
| f(ok(X)) | → | ok(f(X)) | | c(ok(X)) | → | ok(c(X)) |
| g(ok(X)) | → | ok(g(X)) | | d(ok(X)) | → | ok(d(X)) |
| h(ok(X)) | → | ok(h(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: f, g, d, c, active, mark, ok, proper, h, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| f#(mark(X)) | → | f#(X) | | f#(ok(X)) | → | f#(X) |
Problem 8: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
Rewrite Rules
| active(f(f(X))) | → | mark(c(f(g(f(X))))) | | active(c(X)) | → | mark(d(X)) |
| active(h(X)) | → | mark(c(d(X))) | | active(f(X)) | → | f(active(X)) |
| active(h(X)) | → | h(active(X)) | | f(mark(X)) | → | mark(f(X)) |
| h(mark(X)) | → | mark(h(X)) | | proper(f(X)) | → | f(proper(X)) |
| proper(c(X)) | → | c(proper(X)) | | proper(g(X)) | → | g(proper(X)) |
| proper(d(X)) | → | d(proper(X)) | | proper(h(X)) | → | h(proper(X)) |
| f(ok(X)) | → | ok(f(X)) | | c(ok(X)) | → | ok(c(X)) |
| g(ok(X)) | → | ok(g(X)) | | d(ok(X)) | → | ok(d(X)) |
| h(ok(X)) | → | ok(h(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: f, g, d, c, active, mark, ok, proper, h, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
Problem 9: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| top#(mark(X)) | → | top#(proper(X)) | | top#(ok(X)) | → | top#(active(X)) |
Rewrite Rules
| active(f(f(X))) | → | mark(c(f(g(f(X))))) | | active(c(X)) | → | mark(d(X)) |
| active(h(X)) | → | mark(c(d(X))) | | active(f(X)) | → | f(active(X)) |
| active(h(X)) | → | h(active(X)) | | f(mark(X)) | → | mark(f(X)) |
| h(mark(X)) | → | mark(h(X)) | | proper(f(X)) | → | f(proper(X)) |
| proper(c(X)) | → | c(proper(X)) | | proper(g(X)) | → | g(proper(X)) |
| proper(d(X)) | → | d(proper(X)) | | proper(h(X)) | → | h(proper(X)) |
| f(ok(X)) | → | ok(f(X)) | | c(ok(X)) | → | ok(c(X)) |
| g(ok(X)) | → | ok(g(X)) | | d(ok(X)) | → | ok(d(X)) |
| h(ok(X)) | → | ok(h(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: f, g, d, c, active, mark, ok, proper, h, top
Strategy
Polynomial Interpretation
- active(x): 2
- c(x): x
- d(x): x
- f(x): x
- g(x): x
- h(x): x
- mark(x): 2
- ok(x): 2
- proper(x): 1
- top(x): 0
- top#(x): x + 1
Improved Usable rules
| h(ok(X)) | → | ok(h(X)) | | g(ok(X)) | → | ok(g(X)) |
| h(mark(X)) | → | mark(h(X)) | | proper(c(X)) | → | c(proper(X)) |
| c(ok(X)) | → | ok(c(X)) | | d(ok(X)) | → | ok(d(X)) |
| proper(f(X)) | → | f(proper(X)) | | f(ok(X)) | → | ok(f(X)) |
| active(f(X)) | → | f(active(X)) | | active(c(X)) | → | mark(d(X)) |
| proper(g(X)) | → | g(proper(X)) | | active(f(f(X))) | → | mark(c(f(g(f(X))))) |
| active(h(X)) | → | h(active(X)) | | active(h(X)) | → | mark(c(d(X))) |
| proper(d(X)) | → | d(proper(X)) | | proper(h(X)) | → | h(proper(X)) |
| f(mark(X)) | → | mark(f(X)) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| top#(mark(X)) | → | top#(proper(X)) |
Problem 10: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| top#(ok(X)) | → | top#(active(X)) |
Rewrite Rules
| active(f(f(X))) | → | mark(c(f(g(f(X))))) | | active(c(X)) | → | mark(d(X)) |
| active(h(X)) | → | mark(c(d(X))) | | active(f(X)) | → | f(active(X)) |
| active(h(X)) | → | h(active(X)) | | f(mark(X)) | → | mark(f(X)) |
| h(mark(X)) | → | mark(h(X)) | | proper(f(X)) | → | f(proper(X)) |
| proper(c(X)) | → | c(proper(X)) | | proper(g(X)) | → | g(proper(X)) |
| proper(d(X)) | → | d(proper(X)) | | proper(h(X)) | → | h(proper(X)) |
| f(ok(X)) | → | ok(f(X)) | | c(ok(X)) | → | ok(c(X)) |
| g(ok(X)) | → | ok(g(X)) | | d(ok(X)) | → | ok(d(X)) |
| h(ok(X)) | → | ok(h(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: f, g, d, c, active, ok, mark, proper, h, top
Strategy
Polynomial Interpretation
- active(x): 1
- c(x): 1
- d(x): 1
- f(x): x
- g(x): 3
- h(x): x
- mark(x): 1
- ok(x): 2
- proper(x): 0
- top(x): 0
- top#(x): x + 1
Improved Usable rules
| active(c(X)) | → | mark(d(X)) | | h(ok(X)) | → | ok(h(X)) |
| active(f(f(X))) | → | mark(c(f(g(f(X))))) | | active(h(X)) | → | h(active(X)) |
| h(mark(X)) | → | mark(h(X)) | | active(h(X)) | → | mark(c(d(X))) |
| active(f(X)) | → | f(active(X)) | | f(ok(X)) | → | ok(f(X)) |
| f(mark(X)) | → | mark(f(X)) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| top#(ok(X)) | → | top#(active(X)) |