MAYBE
The TRS could not be proven terminating. The proof attempt took 4650 ms.
The following DP Processors were used
Problem 1 was processed with processor PolynomialLinearRange4iUR (0ms).
| – Problem 2 was processed with processor PolynomialLinearRange4iUR (0ms).
| | – Problem 3 remains open; application of the following processors failed [DependencyGraph (0ms), PolynomialLinearRange4iUR (82ms), DependencyGraph (1ms), PolynomialLinearRange8NegiUR (1555ms), DependencyGraph (1ms), ReductionPairSAT (2366ms), DependencyGraph (1ms), SizeChangePrinciple (4ms)].
The following open problems remain:
Open Dependency Pair Problem 3
Dependency Pairs
| f#(a, f(x, a)) | → | f#(a, f(f(a, a), x)) |
Rewrite Rules
| f(a, f(x, a)) | → | f(f(a, f(f(a, a), x)), a) |
Original Signature
Termination of terms over the following signature is verified: f, a
Problem 1: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| f#(a, f(x, a)) | → | f#(f(a, a), x) | | f#(a, f(x, a)) | → | f#(a, a) |
| f#(a, f(x, a)) | → | f#(f(a, f(f(a, a), x)), a) | | f#(a, f(x, a)) | → | f#(a, f(f(a, a), x)) |
Rewrite Rules
| f(a, f(x, a)) | → | f(f(a, f(f(a, a), x)), a) |
Original Signature
Termination of terms over the following signature is verified: f, a
Strategy
Polynomial Interpretation
- a: 2
- f(x,y): 1
- f#(x,y): x + 1
Improved Usable rules
| f(a, f(x, a)) | → | f(f(a, f(f(a, a), x)), a) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| f#(a, f(x, a)) | → | f#(f(a, a), x) | | f#(a, f(x, a)) | → | f#(f(a, f(f(a, a), x)), a) |
Problem 2: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| f#(a, f(x, a)) | → | f#(a, a) | | f#(a, f(x, a)) | → | f#(a, f(f(a, a), x)) |
Rewrite Rules
| f(a, f(x, a)) | → | f(f(a, f(f(a, a), x)), a) |
Original Signature
Termination of terms over the following signature is verified: f, a
Strategy
Polynomial Interpretation
- a: 0
- f(x,y): 1
- f#(x,y): y
Improved Usable rules
| f(a, f(x, a)) | → | f(f(a, f(f(a, a), x)), a) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| f#(a, f(x, a)) | → | f#(a, a) |