YES
The TRS could be proven terminating. The proof took 1332 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (139ms).
| – Problem 2 was processed with processor SubtermCriterion (9ms).
| – Problem 3 was processed with processor SubtermCriterion (12ms).
| – Problem 4 was processed with processor SubtermCriterion (6ms).
| – Problem 5 was processed with processor PolynomialLinearRange4iUR (530ms).
| | – Problem 8 was processed with processor PolynomialLinearRange4iUR (427ms).
| | | – Problem 9 was processed with processor PolynomialLinearRange4iUR (40ms).
| – Problem 6 was processed with processor SubtermCriterion (1ms).
| – Problem 7 was processed with processor SubtermCriterion (0ms).
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| f#(mark(x)) | → | f#(x) | | top#(found(x)) | → | top#(active(x)) |
| top#(mark(x)) | → | check#(x) | | match#(f(x), f(y)) | → | f#(match(x, y)) |
| top#(active(c)) | → | top#(mark(c)) | | top#(found(x)) | → | active#(x) |
| proper#(f(x)) | → | proper#(x) | | check#(x) | → | match#(f(X), x) |
| check#(x) | → | f#(X) | | active#(f(x)) | → | f#(active(x)) |
| check#(x) | → | start#(match(f(X), x)) | | match#(f(x), f(y)) | → | match#(x, y) |
| proper#(f(x)) | → | f#(proper(x)) | | top#(mark(x)) | → | top#(check(x)) |
| f#(found(x)) | → | f#(x) | | check#(f(x)) | → | check#(x) |
| check#(f(x)) | → | f#(check(x)) | | active#(f(x)) | → | active#(x) |
| match#(X, x) | → | proper#(x) | | f#(ok(x)) | → | f#(x) |
Rewrite Rules
| active(f(x)) | → | mark(x) | | top(active(c)) | → | top(mark(c)) |
| top(mark(x)) | → | top(check(x)) | | check(f(x)) | → | f(check(x)) |
| check(x) | → | start(match(f(X), x)) | | match(f(x), f(y)) | → | f(match(x, y)) |
| match(X, x) | → | proper(x) | | proper(c) | → | ok(c) |
| proper(f(x)) | → | f(proper(x)) | | f(ok(x)) | → | ok(f(x)) |
| start(ok(x)) | → | found(x) | | f(found(x)) | → | found(f(x)) |
| top(found(x)) | → | top(active(x)) | | active(f(x)) | → | f(active(x)) |
| f(mark(x)) | → | mark(f(x)) |
Original Signature
Termination of terms over the following signature is verified: f, check, c, start, active, mark, ok, match, proper, X, top, found
Strategy
The following SCCs where found
| match#(f(x), f(y)) → match#(x, y) |
| f#(mark(x)) → f#(x) | f#(found(x)) → f#(x) |
| f#(ok(x)) → f#(x) |
| proper#(f(x)) → proper#(x) |
| top#(found(x)) → top#(active(x)) | top#(active(c)) → top#(mark(c)) |
| top#(mark(x)) → top#(check(x)) |
| active#(f(x)) → active#(x) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| match#(f(x), f(y)) | → | match#(x, y) |
Rewrite Rules
| active(f(x)) | → | mark(x) | | top(active(c)) | → | top(mark(c)) |
| top(mark(x)) | → | top(check(x)) | | check(f(x)) | → | f(check(x)) |
| check(x) | → | start(match(f(X), x)) | | match(f(x), f(y)) | → | f(match(x, y)) |
| match(X, x) | → | proper(x) | | proper(c) | → | ok(c) |
| proper(f(x)) | → | f(proper(x)) | | f(ok(x)) | → | ok(f(x)) |
| start(ok(x)) | → | found(x) | | f(found(x)) | → | found(f(x)) |
| top(found(x)) | → | top(active(x)) | | active(f(x)) | → | f(active(x)) |
| f(mark(x)) | → | mark(f(x)) |
Original Signature
Termination of terms over the following signature is verified: f, check, c, start, active, mark, ok, match, proper, X, top, found
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| match#(f(x), f(y)) | → | match#(x, y) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| f#(mark(x)) | → | f#(x) | | f#(found(x)) | → | f#(x) |
| f#(ok(x)) | → | f#(x) |
Rewrite Rules
| active(f(x)) | → | mark(x) | | top(active(c)) | → | top(mark(c)) |
| top(mark(x)) | → | top(check(x)) | | check(f(x)) | → | f(check(x)) |
| check(x) | → | start(match(f(X), x)) | | match(f(x), f(y)) | → | f(match(x, y)) |
| match(X, x) | → | proper(x) | | proper(c) | → | ok(c) |
| proper(f(x)) | → | f(proper(x)) | | f(ok(x)) | → | ok(f(x)) |
| start(ok(x)) | → | found(x) | | f(found(x)) | → | found(f(x)) |
| top(found(x)) | → | top(active(x)) | | active(f(x)) | → | f(active(x)) |
| f(mark(x)) | → | mark(f(x)) |
Original Signature
Termination of terms over the following signature is verified: f, check, c, start, active, mark, ok, match, proper, X, top, found
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| f#(mark(x)) | → | f#(x) | | f#(found(x)) | → | f#(x) |
| f#(ok(x)) | → | f#(x) |
Problem 4: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
Rewrite Rules
| active(f(x)) | → | mark(x) | | top(active(c)) | → | top(mark(c)) |
| top(mark(x)) | → | top(check(x)) | | check(f(x)) | → | f(check(x)) |
| check(x) | → | start(match(f(X), x)) | | match(f(x), f(y)) | → | f(match(x, y)) |
| match(X, x) | → | proper(x) | | proper(c) | → | ok(c) |
| proper(f(x)) | → | f(proper(x)) | | f(ok(x)) | → | ok(f(x)) |
| start(ok(x)) | → | found(x) | | f(found(x)) | → | found(f(x)) |
| top(found(x)) | → | top(active(x)) | | active(f(x)) | → | f(active(x)) |
| f(mark(x)) | → | mark(f(x)) |
Original Signature
Termination of terms over the following signature is verified: f, check, c, start, active, mark, ok, match, proper, X, top, found
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
Problem 5: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| top#(found(x)) | → | top#(active(x)) | | top#(active(c)) | → | top#(mark(c)) |
| top#(mark(x)) | → | top#(check(x)) |
Rewrite Rules
| active(f(x)) | → | mark(x) | | top(active(c)) | → | top(mark(c)) |
| top(mark(x)) | → | top(check(x)) | | check(f(x)) | → | f(check(x)) |
| check(x) | → | start(match(f(X), x)) | | match(f(x), f(y)) | → | f(match(x, y)) |
| match(X, x) | → | proper(x) | | proper(c) | → | ok(c) |
| proper(f(x)) | → | f(proper(x)) | | f(ok(x)) | → | ok(f(x)) |
| start(ok(x)) | → | found(x) | | f(found(x)) | → | found(f(x)) |
| top(found(x)) | → | top(active(x)) | | active(f(x)) | → | f(active(x)) |
| f(mark(x)) | → | mark(f(x)) |
Original Signature
Termination of terms over the following signature is verified: f, check, c, start, active, mark, ok, match, proper, X, top, found
Strategy
Polynomial Interpretation
- X: 1
- active(x): x
- c: 1
- check(x): 0
- f(x): 0
- found(x): x
- mark(x): 0
- match(x,y): x
- ok(x): x
- proper(x): 1
- start(x): 2x
- top(x): 0
- top#(x): x + 1
Improved Usable rules
| match(X, x) | → | proper(x) | | start(ok(x)) | → | found(x) |
| match(f(x), f(y)) | → | f(match(x, y)) | | proper(f(x)) | → | f(proper(x)) |
| f(mark(x)) | → | mark(f(x)) | | f(found(x)) | → | found(f(x)) |
| f(ok(x)) | → | ok(f(x)) | | proper(c) | → | ok(c) |
| active(f(x)) | → | f(active(x)) | | check(f(x)) | → | f(check(x)) |
| active(f(x)) | → | mark(x) | | check(x) | → | start(match(f(X), x)) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| top#(active(c)) | → | top#(mark(c)) |
Problem 8: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| top#(found(x)) | → | top#(active(x)) | | top#(mark(x)) | → | top#(check(x)) |
Rewrite Rules
| active(f(x)) | → | mark(x) | | top(active(c)) | → | top(mark(c)) |
| top(mark(x)) | → | top(check(x)) | | check(f(x)) | → | f(check(x)) |
| check(x) | → | start(match(f(X), x)) | | match(f(x), f(y)) | → | f(match(x, y)) |
| match(X, x) | → | proper(x) | | proper(c) | → | ok(c) |
| proper(f(x)) | → | f(proper(x)) | | f(ok(x)) | → | ok(f(x)) |
| start(ok(x)) | → | found(x) | | f(found(x)) | → | found(f(x)) |
| top(found(x)) | → | top(active(x)) | | active(f(x)) | → | f(active(x)) |
| f(mark(x)) | → | mark(f(x)) |
Original Signature
Termination of terms over the following signature is verified: f, check, start, c, active, ok, mark, proper, match, X, found, top
Strategy
Polynomial Interpretation
- X: 0
- active(x): x
- c: 0
- check(x): x
- f(x): x + 2
- found(x): x
- mark(x): x + 1
- match(x,y): y
- ok(x): x
- proper(x): x
- start(x): x
- top(x): 0
- top#(x): 2x
Improved Usable rules
| match(X, x) | → | proper(x) | | start(ok(x)) | → | found(x) |
| match(f(x), f(y)) | → | f(match(x, y)) | | proper(f(x)) | → | f(proper(x)) |
| f(mark(x)) | → | mark(f(x)) | | f(found(x)) | → | found(f(x)) |
| f(ok(x)) | → | ok(f(x)) | | proper(c) | → | ok(c) |
| active(f(x)) | → | f(active(x)) | | check(f(x)) | → | f(check(x)) |
| active(f(x)) | → | mark(x) | | check(x) | → | start(match(f(X), x)) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| top#(mark(x)) | → | top#(check(x)) |
Problem 9: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| top#(found(x)) | → | top#(active(x)) |
Rewrite Rules
| active(f(x)) | → | mark(x) | | top(active(c)) | → | top(mark(c)) |
| top(mark(x)) | → | top(check(x)) | | check(f(x)) | → | f(check(x)) |
| check(x) | → | start(match(f(X), x)) | | match(f(x), f(y)) | → | f(match(x, y)) |
| match(X, x) | → | proper(x) | | proper(c) | → | ok(c) |
| proper(f(x)) | → | f(proper(x)) | | f(ok(x)) | → | ok(f(x)) |
| start(ok(x)) | → | found(x) | | f(found(x)) | → | found(f(x)) |
| top(found(x)) | → | top(active(x)) | | active(f(x)) | → | f(active(x)) |
| f(mark(x)) | → | mark(f(x)) |
Original Signature
Termination of terms over the following signature is verified: f, check, c, start, active, mark, ok, match, proper, X, top, found
Strategy
Polynomial Interpretation
- X: 0
- active(x): 0
- c: 0
- check(x): 0
- f(x): 2x
- found(x): 1
- mark(x): 0
- match(x,y): 0
- ok(x): 0
- proper(x): 0
- start(x): 0
- top(x): 0
- top#(x): 2x
Improved Usable rules
| f(mark(x)) | → | mark(f(x)) | | f(found(x)) | → | found(f(x)) |
| f(ok(x)) | → | ok(f(x)) | | active(f(x)) | → | f(active(x)) |
| active(f(x)) | → | mark(x) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| top#(found(x)) | → | top#(active(x)) |
Problem 6: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| active#(f(x)) | → | active#(x) |
Rewrite Rules
| active(f(x)) | → | mark(x) | | top(active(c)) | → | top(mark(c)) |
| top(mark(x)) | → | top(check(x)) | | check(f(x)) | → | f(check(x)) |
| check(x) | → | start(match(f(X), x)) | | match(f(x), f(y)) | → | f(match(x, y)) |
| match(X, x) | → | proper(x) | | proper(c) | → | ok(c) |
| proper(f(x)) | → | f(proper(x)) | | f(ok(x)) | → | ok(f(x)) |
| start(ok(x)) | → | found(x) | | f(found(x)) | → | found(f(x)) |
| top(found(x)) | → | top(active(x)) | | active(f(x)) | → | f(active(x)) |
| f(mark(x)) | → | mark(f(x)) |
Original Signature
Termination of terms over the following signature is verified: f, check, c, start, active, mark, ok, match, proper, X, top, found
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| active#(f(x)) | → | active#(x) |
Problem 7: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| proper#(f(x)) | → | proper#(x) |
Rewrite Rules
| active(f(x)) | → | mark(x) | | top(active(c)) | → | top(mark(c)) |
| top(mark(x)) | → | top(check(x)) | | check(f(x)) | → | f(check(x)) |
| check(x) | → | start(match(f(X), x)) | | match(f(x), f(y)) | → | f(match(x, y)) |
| match(X, x) | → | proper(x) | | proper(c) | → | ok(c) |
| proper(f(x)) | → | f(proper(x)) | | f(ok(x)) | → | ok(f(x)) |
| start(ok(x)) | → | found(x) | | f(found(x)) | → | found(f(x)) |
| top(found(x)) | → | top(active(x)) | | active(f(x)) | → | f(active(x)) |
| f(mark(x)) | → | mark(f(x)) |
Original Signature
Termination of terms over the following signature is verified: f, check, c, start, active, mark, ok, match, proper, X, top, found
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| proper#(f(x)) | → | proper#(x) |