YES
The TRS could be proven terminating. The proof took 1460 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (147ms).
| – Problem 2 was processed with processor SubtermCriterion (1ms).
| – Problem 3 was processed with processor SubtermCriterion (0ms).
| – Problem 4 was processed with processor SubtermCriterion (1ms).
| – Problem 5 was processed with processor SubtermCriterion (0ms).
| – Problem 6 was processed with processor PolynomialLinearRange4iUR (586ms).
| | – Problem 8 was processed with processor PolynomialLinearRange4iUR (422ms).
| – Problem 7 was processed with processor SubtermCriterion (1ms).
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| proper#(f(X)) | → | f#(proper(X)) | | g#(mark(X)) | → | g#(X) |
| top#(mark(X)) | → | top#(proper(X)) | | active#(g(X)) | → | active#(X) |
| proper#(f(X)) | → | proper#(X) | | top#(ok(X)) | → | top#(active(X)) |
| g#(ok(X)) | → | g#(X) | | proper#(c(X)) | → | proper#(X) |
| top#(ok(X)) | → | active#(X) | | c#(ok(X)) | → | c#(X) |
| active#(f(f(a))) | → | c#(f(g(f(a)))) | | active#(f(X)) | → | f#(active(X)) |
| active#(f(f(a))) | → | f#(a) | | active#(f(f(a))) | → | f#(g(f(a))) |
| proper#(g(X)) | → | g#(proper(X)) | | active#(g(X)) | → | g#(active(X)) |
| active#(f(f(a))) | → | g#(f(a)) | | proper#(g(X)) | → | proper#(X) |
| proper#(c(X)) | → | c#(proper(X)) | | f#(mark(X)) | → | f#(X) |
| active#(f(X)) | → | active#(X) | | top#(mark(X)) | → | proper#(X) |
| f#(ok(X)) | → | f#(X) |
Rewrite Rules
| active(f(f(a))) | → | mark(c(f(g(f(a))))) | | active(f(X)) | → | f(active(X)) |
| active(g(X)) | → | g(active(X)) | | f(mark(X)) | → | mark(f(X)) |
| g(mark(X)) | → | mark(g(X)) | | proper(f(X)) | → | f(proper(X)) |
| proper(a) | → | ok(a) | | proper(c(X)) | → | c(proper(X)) |
| proper(g(X)) | → | g(proper(X)) | | f(ok(X)) | → | ok(f(X)) |
| c(ok(X)) | → | ok(c(X)) | | g(ok(X)) | → | ok(g(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, c, a, active, mark, ok, proper, top
Strategy
The following SCCs where found
| f#(mark(X)) → f#(X) | f#(ok(X)) → f#(X) |
| proper#(f(X)) → proper#(X) | proper#(c(X)) → proper#(X) |
| proper#(g(X)) → proper#(X) |
| g#(mark(X)) → g#(X) | g#(ok(X)) → g#(X) |
| active#(g(X)) → active#(X) | active#(f(X)) → active#(X) |
| top#(mark(X)) → top#(proper(X)) | top#(ok(X)) → top#(active(X)) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| active#(g(X)) | → | active#(X) | | active#(f(X)) | → | active#(X) |
Rewrite Rules
| active(f(f(a))) | → | mark(c(f(g(f(a))))) | | active(f(X)) | → | f(active(X)) |
| active(g(X)) | → | g(active(X)) | | f(mark(X)) | → | mark(f(X)) |
| g(mark(X)) | → | mark(g(X)) | | proper(f(X)) | → | f(proper(X)) |
| proper(a) | → | ok(a) | | proper(c(X)) | → | c(proper(X)) |
| proper(g(X)) | → | g(proper(X)) | | f(ok(X)) | → | ok(f(X)) |
| c(ok(X)) | → | ok(c(X)) | | g(ok(X)) | → | ok(g(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, c, a, active, mark, ok, proper, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| active#(g(X)) | → | active#(X) | | active#(f(X)) | → | active#(X) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| g#(mark(X)) | → | g#(X) | | g#(ok(X)) | → | g#(X) |
Rewrite Rules
| active(f(f(a))) | → | mark(c(f(g(f(a))))) | | active(f(X)) | → | f(active(X)) |
| active(g(X)) | → | g(active(X)) | | f(mark(X)) | → | mark(f(X)) |
| g(mark(X)) | → | mark(g(X)) | | proper(f(X)) | → | f(proper(X)) |
| proper(a) | → | ok(a) | | proper(c(X)) | → | c(proper(X)) |
| proper(g(X)) | → | g(proper(X)) | | f(ok(X)) | → | ok(f(X)) |
| c(ok(X)) | → | ok(c(X)) | | g(ok(X)) | → | ok(g(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, c, a, active, mark, ok, proper, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| g#(mark(X)) | → | g#(X) | | g#(ok(X)) | → | g#(X) |
Problem 4: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| f#(mark(X)) | → | f#(X) | | f#(ok(X)) | → | f#(X) |
Rewrite Rules
| active(f(f(a))) | → | mark(c(f(g(f(a))))) | | active(f(X)) | → | f(active(X)) |
| active(g(X)) | → | g(active(X)) | | f(mark(X)) | → | mark(f(X)) |
| g(mark(X)) | → | mark(g(X)) | | proper(f(X)) | → | f(proper(X)) |
| proper(a) | → | ok(a) | | proper(c(X)) | → | c(proper(X)) |
| proper(g(X)) | → | g(proper(X)) | | f(ok(X)) | → | ok(f(X)) |
| c(ok(X)) | → | ok(c(X)) | | g(ok(X)) | → | ok(g(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, c, a, active, mark, ok, proper, 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 5: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| proper#(f(X)) | → | proper#(X) | | proper#(c(X)) | → | proper#(X) |
| proper#(g(X)) | → | proper#(X) |
Rewrite Rules
| active(f(f(a))) | → | mark(c(f(g(f(a))))) | | active(f(X)) | → | f(active(X)) |
| active(g(X)) | → | g(active(X)) | | f(mark(X)) | → | mark(f(X)) |
| g(mark(X)) | → | mark(g(X)) | | proper(f(X)) | → | f(proper(X)) |
| proper(a) | → | ok(a) | | proper(c(X)) | → | c(proper(X)) |
| proper(g(X)) | → | g(proper(X)) | | f(ok(X)) | → | ok(f(X)) |
| c(ok(X)) | → | ok(c(X)) | | g(ok(X)) | → | ok(g(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, c, a, active, mark, ok, proper, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| proper#(f(X)) | → | proper#(X) | | proper#(c(X)) | → | proper#(X) |
| proper#(g(X)) | → | proper#(X) |
Problem 6: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| top#(mark(X)) | → | top#(proper(X)) | | top#(ok(X)) | → | top#(active(X)) |
Rewrite Rules
| active(f(f(a))) | → | mark(c(f(g(f(a))))) | | active(f(X)) | → | f(active(X)) |
| active(g(X)) | → | g(active(X)) | | f(mark(X)) | → | mark(f(X)) |
| g(mark(X)) | → | mark(g(X)) | | proper(f(X)) | → | f(proper(X)) |
| proper(a) | → | ok(a) | | proper(c(X)) | → | c(proper(X)) |
| proper(g(X)) | → | g(proper(X)) | | f(ok(X)) | → | ok(f(X)) |
| c(ok(X)) | → | ok(c(X)) | | g(ok(X)) | → | ok(g(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, c, a, active, mark, ok, proper, top
Strategy
Polynomial Interpretation
- a: 0
- active(x): x
- c(x): 0
- f(x): 2x + 1
- g(x): x
- mark(x): x + 1
- ok(x): x
- proper(x): x
- top(x): 0
- top#(x): x
Improved Usable rules
| proper(g(X)) | → | g(proper(X)) | | g(ok(X)) | → | ok(g(X)) |
| proper(c(X)) | → | c(proper(X)) | | proper(a) | → | ok(a) |
| active(g(X)) | → | g(active(X)) | | c(ok(X)) | → | ok(c(X)) |
| active(f(f(a))) | → | mark(c(f(g(f(a))))) | | proper(f(X)) | → | f(proper(X)) |
| g(mark(X)) | → | mark(g(X)) | | f(ok(X)) | → | ok(f(X)) |
| f(mark(X)) | → | mark(f(X)) | | active(f(X)) | → | f(active(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 8: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| top#(ok(X)) | → | top#(active(X)) |
Rewrite Rules
| active(f(f(a))) | → | mark(c(f(g(f(a))))) | | active(f(X)) | → | f(active(X)) |
| active(g(X)) | → | g(active(X)) | | f(mark(X)) | → | mark(f(X)) |
| g(mark(X)) | → | mark(g(X)) | | proper(f(X)) | → | f(proper(X)) |
| proper(a) | → | ok(a) | | proper(c(X)) | → | c(proper(X)) |
| proper(g(X)) | → | g(proper(X)) | | f(ok(X)) | → | ok(f(X)) |
| c(ok(X)) | → | ok(c(X)) | | g(ok(X)) | → | ok(g(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, c, a, active, ok, mark, proper, top
Strategy
Polynomial Interpretation
- a: 0
- active(x): 0
- c(x): 2
- f(x): 2x
- g(x): 2x
- mark(x): 0
- ok(x): 1
- proper(x): 0
- top(x): 0
- top#(x): x + 1
Improved Usable rules
| g(ok(X)) | → | ok(g(X)) | | active(g(X)) | → | g(active(X)) |
| active(f(f(a))) | → | mark(c(f(g(f(a))))) | | g(mark(X)) | → | mark(g(X)) |
| active(f(X)) | → | f(active(X)) | | f(mark(X)) | → | mark(f(X)) |
| f(ok(X)) | → | ok(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)) |
Problem 7: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
Rewrite Rules
| active(f(f(a))) | → | mark(c(f(g(f(a))))) | | active(f(X)) | → | f(active(X)) |
| active(g(X)) | → | g(active(X)) | | f(mark(X)) | → | mark(f(X)) |
| g(mark(X)) | → | mark(g(X)) | | proper(f(X)) | → | f(proper(X)) |
| proper(a) | → | ok(a) | | proper(c(X)) | → | c(proper(X)) |
| proper(g(X)) | → | g(proper(X)) | | f(ok(X)) | → | ok(f(X)) |
| c(ok(X)) | → | ok(c(X)) | | g(ok(X)) | → | ok(g(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, c, a, active, mark, ok, proper, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed: