MAYBE
The TRS could not be proven terminating. The proof attempt took 12329 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 (1ms), PolynomialLinearRange4iUR (185ms), DependencyGraph (2ms), PolynomialLinearRange8NegiUR (7034ms), DependencyGraph (1ms), ReductionPairSAT (4343ms), DependencyGraph (2ms), SizeChangePrinciple (5ms)].
The following open problems remain:
Open Dependency Pair Problem 3
Dependency Pairs
| f#(f(a, x), a) | → | f#(f(x, f(a, a)), a) | | f#(f(a, x), a) | → | f#(f(f(x, f(a, a)), a), a) |
Rewrite Rules
| f(f(a, x), a) | → | f(f(f(x, f(a, a)), 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, x), a) | → | f#(f(x, f(a, a)), a) | | f#(f(a, x), a) | → | f#(a, a) |
| f#(f(a, x), a) | → | f#(x, f(a, a)) | | f#(f(a, x), a) | → | f#(f(f(x, f(a, a)), a), a) |
Rewrite Rules
| f(f(a, x), a) | → | f(f(f(x, f(a, a)), a), a) |
Original Signature
Termination of terms over the following signature is verified: f, a
Strategy
Polynomial Interpretation
- a: 1
- f(x,y): 0
- f#(x,y): y
Improved Usable rules
| f(f(a, x), a) | → | f(f(f(x, f(a, a)), a), a) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| f#(f(a, x), a) | → | f#(x, f(a, a)) |
Problem 2: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| f#(f(a, x), a) | → | f#(f(x, f(a, a)), a) | | f#(f(a, x), a) | → | f#(a, a) |
| f#(f(a, x), a) | → | f#(f(f(x, f(a, a)), a), a) |
Rewrite Rules
| f(f(a, x), a) | → | f(f(f(x, f(a, a)), a), 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): x
Improved Usable rules
| f(f(a, x), a) | → | f(f(f(x, f(a, a)), a), a) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| f#(f(a, x), a) | → | f#(a, a) |