MAYBE
The TRS could not be proven terminating. The proof attempt took 18018 ms.
The following DP Processors were used
Problem 1 was processed with processor PolynomialLinearRange4iUR (0ms).
| – Problem 2 was processed with processor DependencyGraph (0ms).
| | – Problem 3 remains open; application of the following processors failed [PolynomialLinearRange4iUR (769ms), DependencyGraph (1ms), PolynomialLinearRange8NegiUR (9405ms), DependencyGraph (1ms), ReductionPairSAT (6251ms), DependencyGraph (1ms), SizeChangePrinciple (26ms)].
The following open problems remain:
Open Dependency Pair Problem 3
Dependency Pairs
| +#(+(x, *(y, z)), *(y, u)) | → | +#(x, *(y, +(z, u))) | | +#(x, +(y, z)) | → | +#(+(x, y), z) |
Rewrite Rules
| +(x, +(y, z)) | → | +(+(x, y), z) | | *(x, +(y, z)) | → | +(*(x, y), *(x, z)) |
| +(+(x, *(y, z)), *(y, u)) | → | +(x, *(y, +(z, u))) |
Original Signature
Termination of terms over the following signature is verified: *, +
Problem 1: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| *#(x, +(y, z)) | → | *#(x, z) | | *#(x, +(y, z)) | → | +#(*(x, y), *(x, z)) |
| +#(+(x, *(y, z)), *(y, u)) | → | *#(y, +(z, u)) | | +#(+(x, *(y, z)), *(y, u)) | → | +#(x, *(y, +(z, u))) |
| +#(x, +(y, z)) | → | +#(+(x, y), z) | | +#(+(x, *(y, z)), *(y, u)) | → | +#(z, u) |
| *#(x, +(y, z)) | → | *#(x, y) | | +#(x, +(y, z)) | → | +#(x, y) |
Rewrite Rules
| +(x, +(y, z)) | → | +(+(x, y), z) | | *(x, +(y, z)) | → | +(*(x, y), *(x, z)) |
| +(+(x, *(y, z)), *(y, u)) | → | +(x, *(y, +(z, u))) |
Original Signature
Termination of terms over the following signature is verified: *, +
Strategy
Polynomial Interpretation
- *(x,y): y
- *#(x,y): 2y
- +(x,y): y + x + 1
- +#(x,y): 2y + 2x
Improved Usable rules
| +(x, +(y, z)) | → | +(+(x, y), z) | | *(x, +(y, z)) | → | +(*(x, y), *(x, z)) |
| +(+(x, *(y, z)), *(y, u)) | → | +(x, *(y, +(z, u))) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| *#(x, +(y, z)) | → | *#(x, z) | | *#(x, +(y, z)) | → | +#(*(x, y), *(x, z)) |
| +#(+(x, *(y, z)), *(y, u)) | → | +#(z, u) | | *#(x, +(y, z)) | → | *#(x, y) |
| +#(x, +(y, z)) | → | +#(x, y) |
Problem 2: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| +#(+(x, *(y, z)), *(y, u)) | → | *#(y, +(z, u)) | | +#(+(x, *(y, z)), *(y, u)) | → | +#(x, *(y, +(z, u))) |
| +#(x, +(y, z)) | → | +#(+(x, y), z) |
Rewrite Rules
| +(x, +(y, z)) | → | +(+(x, y), z) | | *(x, +(y, z)) | → | +(*(x, y), *(x, z)) |
| +(+(x, *(y, z)), *(y, u)) | → | +(x, *(y, +(z, u))) |
Original Signature
Termination of terms over the following signature is verified: *, +
Strategy
The following SCCs where found
| +#(+(x, *(y, z)), *(y, u)) → +#(x, *(y, +(z, u))) | +#(x, +(y, z)) → +#(+(x, y), z) |