YES
The TRS could be proven terminating. The proof took 981 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (127ms).
| – Problem 2 was processed with processor SubtermCriterion (2ms).
| – Problem 3 was processed with processor SubtermCriterion (40ms).
| – Problem 4 was processed with processor PolynomialLinearRange4iUR (518ms).
| | – Problem 7 was processed with processor PolynomialLinearRange4iUR (135ms).
| – Problem 5 was processed with processor SubtermCriterion (8ms).
| – Problem 6 was processed with processor SubtermCriterion (1ms).
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| active#(f(g(X), Y)) | → | g#(X) | | active#(f(X1, X2)) | → | f#(active(X1), X2) |
| g#(mark(X)) | → | g#(X) | | top#(mark(X)) | → | top#(proper(X)) |
| active#(f(g(X), Y)) | → | f#(X, f(g(X), Y)) | | active#(g(X)) | → | active#(X) |
| top#(ok(X)) | → | top#(active(X)) | | proper#(f(X1, X2)) | → | proper#(X2) |
| g#(ok(X)) | → | g#(X) | | proper#(f(X1, X2)) | → | proper#(X1) |
| proper#(f(X1, X2)) | → | f#(proper(X1), proper(X2)) | | top#(ok(X)) | → | active#(X) |
| proper#(g(X)) | → | g#(proper(X)) | | active#(g(X)) | → | g#(active(X)) |
| proper#(g(X)) | → | proper#(X) | | f#(ok(X1), ok(X2)) | → | f#(X1, X2) |
| active#(f(X1, X2)) | → | active#(X1) | | f#(mark(X1), X2) | → | f#(X1, X2) |
| active#(f(g(X), Y)) | → | f#(g(X), Y) | | top#(mark(X)) | → | proper#(X) |
Rewrite Rules
| active(f(g(X), Y)) | → | mark(f(X, f(g(X), Y))) | | active(f(X1, X2)) | → | f(active(X1), X2) |
| active(g(X)) | → | g(active(X)) | | f(mark(X1), X2) | → | mark(f(X1, X2)) |
| g(mark(X)) | → | mark(g(X)) | | proper(f(X1, X2)) | → | f(proper(X1), proper(X2)) |
| proper(g(X)) | → | g(proper(X)) | | f(ok(X1), ok(X2)) | → | ok(f(X1, X2)) |
| 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, active, mark, ok, proper, top
Strategy
The following SCCs where found
| proper#(f(X1, X2)) → proper#(X2) | proper#(f(X1, X2)) → proper#(X1) |
| proper#(g(X)) → proper#(X) |
| f#(ok(X1), ok(X2)) → f#(X1, X2) | f#(mark(X1), X2) → f#(X1, X2) |
| active#(g(X)) → active#(X) | active#(f(X1, X2)) → active#(X1) |
| g#(mark(X)) → g#(X) | g#(ok(X)) → g#(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(X1, X2)) | → | active#(X1) |
Rewrite Rules
| active(f(g(X), Y)) | → | mark(f(X, f(g(X), Y))) | | active(f(X1, X2)) | → | f(active(X1), X2) |
| active(g(X)) | → | g(active(X)) | | f(mark(X1), X2) | → | mark(f(X1, X2)) |
| g(mark(X)) | → | mark(g(X)) | | proper(f(X1, X2)) | → | f(proper(X1), proper(X2)) |
| proper(g(X)) | → | g(proper(X)) | | f(ok(X1), ok(X2)) | → | ok(f(X1, X2)) |
| 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, 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(X1, X2)) | → | active#(X1) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| f#(ok(X1), ok(X2)) | → | f#(X1, X2) | | f#(mark(X1), X2) | → | f#(X1, X2) |
Rewrite Rules
| active(f(g(X), Y)) | → | mark(f(X, f(g(X), Y))) | | active(f(X1, X2)) | → | f(active(X1), X2) |
| active(g(X)) | → | g(active(X)) | | f(mark(X1), X2) | → | mark(f(X1, X2)) |
| g(mark(X)) | → | mark(g(X)) | | proper(f(X1, X2)) | → | f(proper(X1), proper(X2)) |
| proper(g(X)) | → | g(proper(X)) | | f(ok(X1), ok(X2)) | → | ok(f(X1, X2)) |
| 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, active, mark, ok, proper, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| f#(ok(X1), ok(X2)) | → | f#(X1, X2) | | f#(mark(X1), X2) | → | f#(X1, X2) |
Problem 4: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| top#(mark(X)) | → | top#(proper(X)) | | top#(ok(X)) | → | top#(active(X)) |
Rewrite Rules
| active(f(g(X), Y)) | → | mark(f(X, f(g(X), Y))) | | active(f(X1, X2)) | → | f(active(X1), X2) |
| active(g(X)) | → | g(active(X)) | | f(mark(X1), X2) | → | mark(f(X1, X2)) |
| g(mark(X)) | → | mark(g(X)) | | proper(f(X1, X2)) | → | f(proper(X1), proper(X2)) |
| proper(g(X)) | → | g(proper(X)) | | f(ok(X1), ok(X2)) | → | ok(f(X1, X2)) |
| 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, active, mark, ok, proper, top
Strategy
Polynomial Interpretation
- active(x): x
- f(x,y): 2x
- g(x): 3x
- mark(x): 2x
- ok(x): x + 1
- proper(x): x
- top(x): 0
- top#(x): 2x + 1
Improved Usable rules
| proper(g(X)) | → | g(proper(X)) | | g(ok(X)) | → | ok(g(X)) |
| proper(f(X1, X2)) | → | f(proper(X1), proper(X2)) | | active(f(g(X), Y)) | → | mark(f(X, f(g(X), Y))) |
| active(g(X)) | → | g(active(X)) | | f(ok(X1), ok(X2)) | → | ok(f(X1, X2)) |
| f(mark(X1), X2) | → | mark(f(X1, X2)) | | g(mark(X)) | → | mark(g(X)) |
| active(f(X1, X2)) | → | f(active(X1), X2) |
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: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| top#(mark(X)) | → | top#(proper(X)) |
Rewrite Rules
| active(f(g(X), Y)) | → | mark(f(X, f(g(X), Y))) | | active(f(X1, X2)) | → | f(active(X1), X2) |
| active(g(X)) | → | g(active(X)) | | f(mark(X1), X2) | → | mark(f(X1, X2)) |
| g(mark(X)) | → | mark(g(X)) | | proper(f(X1, X2)) | → | f(proper(X1), proper(X2)) |
| proper(g(X)) | → | g(proper(X)) | | f(ok(X1), ok(X2)) | → | ok(f(X1, X2)) |
| 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, active, ok, mark, proper, top
Strategy
Polynomial Interpretation
- active(x): 0
- f(x,y): x
- g(x): 2x
- mark(x): 1
- ok(x): 0
- proper(x): 0
- top(x): 0
- top#(x): 2x + 1
Improved Usable rules
| proper(g(X)) | → | g(proper(X)) | | g(ok(X)) | → | ok(g(X)) |
| proper(f(X1, X2)) | → | f(proper(X1), proper(X2)) | | f(ok(X1), ok(X2)) | → | ok(f(X1, X2)) |
| f(mark(X1), X2) | → | mark(f(X1, X2)) | | g(mark(X)) | → | mark(g(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 5: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| proper#(f(X1, X2)) | → | proper#(X2) | | proper#(f(X1, X2)) | → | proper#(X1) |
| proper#(g(X)) | → | proper#(X) |
Rewrite Rules
| active(f(g(X), Y)) | → | mark(f(X, f(g(X), Y))) | | active(f(X1, X2)) | → | f(active(X1), X2) |
| active(g(X)) | → | g(active(X)) | | f(mark(X1), X2) | → | mark(f(X1, X2)) |
| g(mark(X)) | → | mark(g(X)) | | proper(f(X1, X2)) | → | f(proper(X1), proper(X2)) |
| proper(g(X)) | → | g(proper(X)) | | f(ok(X1), ok(X2)) | → | ok(f(X1, X2)) |
| 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, active, mark, ok, proper, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| proper#(f(X1, X2)) | → | proper#(X2) | | proper#(f(X1, X2)) | → | proper#(X1) |
| proper#(g(X)) | → | proper#(X) |
Problem 6: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| g#(mark(X)) | → | g#(X) | | g#(ok(X)) | → | g#(X) |
Rewrite Rules
| active(f(g(X), Y)) | → | mark(f(X, f(g(X), Y))) | | active(f(X1, X2)) | → | f(active(X1), X2) |
| active(g(X)) | → | g(active(X)) | | f(mark(X1), X2) | → | mark(f(X1, X2)) |
| g(mark(X)) | → | mark(g(X)) | | proper(f(X1, X2)) | → | f(proper(X1), proper(X2)) |
| proper(g(X)) | → | g(proper(X)) | | f(ok(X1), ok(X2)) | → | ok(f(X1, X2)) |
| 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, 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) |