MAYBE
The TRS could not be proven terminating. The proof attempt took 9293 ms.
The following DP Processors were used
Problem 1 was processed with processor PolynomialLinearRange4iUR (0ms).
| – Problem 2 remains open; application of the following processors failed [DependencyGraph (5ms), PolynomialLinearRange4iUR (214ms), DependencyGraph (3ms), PolynomialLinearRange8NegiUR (3987ms), DependencyGraph (3ms), ReductionPairSAT (4488ms), DependencyGraph (2ms), SizeChangePrinciple (25ms)].
The following open problems remain:
Open Dependency Pair Problem 2
Dependency Pairs
| f#(f(a, a), x) | → | f#(a, f(a, a)) | | f#(f(a, a), x) | → | f#(x, f(a, f(a, a))) |
| f#(f(a, a), x) | → | f#(a, f(x, f(a, f(a, a)))) |
Rewrite Rules
| f(f(a, a), x) | → | f(a, f(x, f(a, f(a, a)))) |
Original Signature
Termination of terms over the following signature is verified: f, a
Problem 1: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| f#(f(a, a), x) | → | f#(a, f(a, a)) | | f#(f(a, a), x) | → | f#(a, a) |
| f#(f(a, a), x) | → | f#(x, f(a, f(a, a))) | | f#(f(a, a), x) | → | f#(a, f(x, f(a, f(a, a)))) |
Rewrite Rules
| f(f(a, a), x) | → | f(a, f(x, f(a, f(a, a)))) |
Original Signature
Termination of terms over the following signature is verified: f, a
Strategy
Polynomial Interpretation
- a: 0
- f(x,y): x + 1
- f#(x,y): y + x
Improved Usable rules
| f(f(a, a), x) | → | f(a, f(x, f(a, f(a, a)))) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| f#(f(a, a), x) | → | f#(a, a) |