TIMEOUT
The TRS could not be proven terminating. The proof attempt took 60119 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (47ms).
| – Problem 2 remains open; application of the following processors failed [SubtermCriterion (1ms), DependencyGraph (5ms), PolynomialLinearRange4iUR (1174ms), DependencyGraph (4ms), PolynomialLinearRange8NegiUR (17733ms), DependencyGraph (2ms), ReductionPairSAT (1155ms), DependencyGraph (1ms), SizeChangePrinciple (171ms), ForwardNarrowing (1ms), BackwardInstantiation (1ms), ForwardInstantiation (1ms), Propagation (0ms)].
| – Problem 3 was processed with processor SubtermCriterion (1ms).
| – Problem 4 was processed with processor SubtermCriterion (1ms).
| – Problem 5 was processed with processor SubtermCriterion (0ms).
The following open problems remain:
Open Dependency Pair Problem 2
Dependency Pairs
| f#(s(x), s(y)) | → | f#(-(y, min(x, y)), s(twice(min(x, y)))) | | f#(s(x), s(y)) | → | f#(-(x, min(x, y)), s(twice(min(x, y)))) |
Rewrite Rules
| -(x, 0) | → | x | | -(s(x), s(y)) | → | -(x, y) |
| min(x, 0) | → | 0 | | min(0, y) | → | 0 |
| min(s(x), s(y)) | → | s(min(x, y)) | | twice(0) | → | 0 |
| twice(s(x)) | → | s(s(twice(x))) | | f(s(x), s(y)) | → | f(-(y, min(x, y)), s(twice(min(x, y)))) |
| f(s(x), s(y)) | → | f(-(x, min(x, y)), s(twice(min(x, y)))) |
Original Signature
Termination of terms over the following signature is verified: f, min, twice, 0, s, -
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| f#(s(x), s(y)) | → | f#(-(y, min(x, y)), s(twice(min(x, y)))) | | f#(s(x), s(y)) | → | f#(-(x, min(x, y)), s(twice(min(x, y)))) |
| f#(s(x), s(y)) | → | -#(x, min(x, y)) | | f#(s(x), s(y)) | → | twice#(min(x, y)) |
| min#(s(x), s(y)) | → | min#(x, y) | | f#(s(x), s(y)) | → | -#(y, min(x, y)) |
| f#(s(x), s(y)) | → | min#(x, y) | | twice#(s(x)) | → | twice#(x) |
| -#(s(x), s(y)) | → | -#(x, y) |
Rewrite Rules
| -(x, 0) | → | x | | -(s(x), s(y)) | → | -(x, y) |
| min(x, 0) | → | 0 | | min(0, y) | → | 0 |
| min(s(x), s(y)) | → | s(min(x, y)) | | twice(0) | → | 0 |
| twice(s(x)) | → | s(s(twice(x))) | | f(s(x), s(y)) | → | f(-(y, min(x, y)), s(twice(min(x, y)))) |
| f(s(x), s(y)) | → | f(-(x, min(x, y)), s(twice(min(x, y)))) |
Original Signature
Termination of terms over the following signature is verified: min, f, 0, twice, s, -
Strategy
The following SCCs where found
| min#(s(x), s(y)) → min#(x, y) |
| -#(s(x), s(y)) → -#(x, y) |
| f#(s(x), s(y)) → f#(-(y, min(x, y)), s(twice(min(x, y)))) | f#(s(x), s(y)) → f#(-(x, min(x, y)), s(twice(min(x, y)))) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| -#(s(x), s(y)) | → | -#(x, y) |
Rewrite Rules
| -(x, 0) | → | x | | -(s(x), s(y)) | → | -(x, y) |
| min(x, 0) | → | 0 | | min(0, y) | → | 0 |
| min(s(x), s(y)) | → | s(min(x, y)) | | twice(0) | → | 0 |
| twice(s(x)) | → | s(s(twice(x))) | | f(s(x), s(y)) | → | f(-(y, min(x, y)), s(twice(min(x, y)))) |
| f(s(x), s(y)) | → | f(-(x, min(x, y)), s(twice(min(x, y)))) |
Original Signature
Termination of terms over the following signature is verified: min, f, 0, twice, s, -
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| -#(s(x), s(y)) | → | -#(x, y) |
Problem 4: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
Rewrite Rules
| -(x, 0) | → | x | | -(s(x), s(y)) | → | -(x, y) |
| min(x, 0) | → | 0 | | min(0, y) | → | 0 |
| min(s(x), s(y)) | → | s(min(x, y)) | | twice(0) | → | 0 |
| twice(s(x)) | → | s(s(twice(x))) | | f(s(x), s(y)) | → | f(-(y, min(x, y)), s(twice(min(x, y)))) |
| f(s(x), s(y)) | → | f(-(x, min(x, y)), s(twice(min(x, y)))) |
Original Signature
Termination of terms over the following signature is verified: min, f, 0, twice, s, -
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
Problem 5: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| min#(s(x), s(y)) | → | min#(x, y) |
Rewrite Rules
| -(x, 0) | → | x | | -(s(x), s(y)) | → | -(x, y) |
| min(x, 0) | → | 0 | | min(0, y) | → | 0 |
| min(s(x), s(y)) | → | s(min(x, y)) | | twice(0) | → | 0 |
| twice(s(x)) | → | s(s(twice(x))) | | f(s(x), s(y)) | → | f(-(y, min(x, y)), s(twice(min(x, y)))) |
| f(s(x), s(y)) | → | f(-(x, min(x, y)), s(twice(min(x, y)))) |
Original Signature
Termination of terms over the following signature is verified: min, f, 0, twice, s, -
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| min#(s(x), s(y)) | → | min#(x, y) |