MAYBE
The TRS could not be proven terminating. The proof attempt took 1784 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 was processed with processor PolynomialLinearRange4iUR (0ms).
| | | – Problem 4 remains open; application of the following processors failed [DependencyGraph (1ms), PolynomialLinearRange4iUR (43ms), DependencyGraph (1ms), PolynomialLinearRange8NegiUR (224ms), DependencyGraph (0ms), ReductionPairSAT (568ms), DependencyGraph (1ms), SizeChangePrinciple (52ms)].
The following open problems remain:
Open Dependency Pair Problem 4
Dependency Pairs
| *#(x, +(y, f(z))) | → | *#(g(x, z), +(y, y)) |
Rewrite Rules
| *(*(x, y), z) | → | *(x, *(y, z)) | | *(+(x, y), z) | → | +(*(x, z), *(y, z)) |
| *(x, +(y, f(z))) | → | *(g(x, z), +(y, y)) |
Original Signature
Termination of terms over the following signature is verified: f, g, *, +
Problem 1: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| *#(+(x, y), z) | → | *#(x, z) | | *#(*(x, y), z) | → | *#(x, *(y, z)) |
| *#(*(x, y), z) | → | *#(y, z) | | *#(+(x, y), z) | → | *#(y, z) |
| *#(x, +(y, f(z))) | → | *#(g(x, z), +(y, y)) |
Rewrite Rules
| *(*(x, y), z) | → | *(x, *(y, z)) | | *(+(x, y), z) | → | +(*(x, z), *(y, z)) |
| *(x, +(y, f(z))) | → | *(g(x, z), +(y, y)) |
Original Signature
Termination of terms over the following signature is verified: f, g, *, +
Strategy
Polynomial Interpretation
- *(x,y): y + x
- *#(x,y): 2x + 1
- +(x,y): y + 2x + 1
- f(x): 1
- g(x,y): x
Improved Usable rules
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) | → | *#(y, z) |
Problem 2: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| *#(*(x, y), z) | → | *#(y, z) | | *#(*(x, y), z) | → | *#(x, *(y, z)) |
| *#(x, +(y, f(z))) | → | *#(g(x, z), +(y, y)) |
Rewrite Rules
| *(*(x, y), z) | → | *(x, *(y, z)) | | *(+(x, y), z) | → | +(*(x, z), *(y, z)) |
| *(x, +(y, f(z))) | → | *(g(x, z), +(y, y)) |
Original Signature
Termination of terms over the following signature is verified: f, g, *, +
Strategy
Polynomial Interpretation
- *(x,y): y + x + 1
- *#(x,y): 2y + 2x
- +(x,y): 0
- f(x): 3x
- g(x,y): x
Improved Usable rules
| *(+(x, y), z) | → | +(*(x, z), *(y, z)) | | *(x, +(y, f(z))) | → | *(g(x, z), +(y, y)) |
| *(*(x, y), z) | → | *(x, *(y, z)) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| *#(*(x, y), z) | → | *#(y, z) |
Problem 3: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| *#(*(x, y), z) | → | *#(x, *(y, z)) | | *#(x, +(y, f(z))) | → | *#(g(x, z), +(y, y)) |
Rewrite Rules
| *(*(x, y), z) | → | *(x, *(y, z)) | | *(+(x, y), z) | → | +(*(x, z), *(y, z)) |
| *(x, +(y, f(z))) | → | *(g(x, z), +(y, y)) |
Original Signature
Termination of terms over the following signature is verified: f, g, *, +
Strategy
Polynomial Interpretation
- *(x,y): 2x + 1
- *#(x,y): 2x
- +(x,y): 2y
- f(x): 1
- g(x,y): x
Improved Usable rules
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| *#(*(x, y), z) | → | *#(x, *(y, z)) |