MAYBE
The TRS could not be proven terminating. The proof attempt took 4552 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 (2ms), PolynomialLinearRange4iUR (212ms), DependencyGraph (3ms), PolynomialLinearRange8NegiUR (1494ms), DependencyGraph (2ms), ReductionPairSAT (1935ms), DependencyGraph (3ms), SizeChangePrinciple (7ms)].
The following open problems remain:
Open Dependency Pair Problem 3
Dependency Pairs
| f#(x, f(y, z)) | → | f#(x, y) | | f#(f(a, a), x) | → | f#(a, f(b, f(a, x))) |
| f#(x, f(y, z)) | → | f#(f(x, y), z) |
Rewrite Rules
| f(f(a, a), x) | → | f(a, f(b, f(a, x))) | | f(x, f(y, z)) | → | f(f(x, y), z) |
Original Signature
Termination of terms over the following signature is verified: f, b, a
Problem 1: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| f#(f(a, a), x) | → | f#(b, f(a, x)) | | f#(x, f(y, z)) | → | f#(x, y) |
| f#(f(a, a), x) | → | f#(a, f(b, f(a, x))) | | f#(x, f(y, z)) | → | f#(f(x, y), z) |
| f#(f(a, a), x) | → | f#(a, x) |
Rewrite Rules
| f(f(a, a), x) | → | f(a, f(b, f(a, x))) | | f(x, f(y, z)) | → | f(f(x, y), z) |
Original Signature
Termination of terms over the following signature is verified: f, b, a
Strategy
Polynomial Interpretation
- a: 1
- b: 0
- f(x,y): x
- f#(x,y): 2x
Improved Usable rules
| f(x, f(y, z)) | → | f(f(x, y), z) | | f(f(a, a), x) | → | f(a, f(b, f(a, x))) |
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#(b, f(a, x)) |
Problem 2: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| f#(x, f(y, z)) | → | f#(x, y) | | f#(f(a, a), x) | → | f#(a, f(b, f(a, x))) |
| f#(x, f(y, z)) | → | f#(f(x, y), z) | | f#(f(a, a), x) | → | f#(a, x) |
Rewrite Rules
| f(f(a, a), x) | → | f(a, f(b, f(a, x))) | | f(x, f(y, z)) | → | f(f(x, y), z) |
Original Signature
Termination of terms over the following signature is verified: f, b, a
Strategy
Polynomial Interpretation
- a: 1
- b: 0
- f(x,y): y + x
- f#(x,y): y + x
Improved Usable rules
| f(x, f(y, z)) | → | f(f(x, y), z) | | f(f(a, a), x) | → | f(a, f(b, f(a, x))) |
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, x) |