TIMEOUT
The TRS could not be proven terminating. The proof attempt took 60026 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (58ms).
| – Problem 2 remains open; application of the following processors failed [SubtermCriterion (0ms), DependencyGraph (1ms), PolynomialLinearRange4iUR (1126ms), DependencyGraph (1ms), PolynomialLinearRange8NegiUR (27635ms), DependencyGraph (1ms), ReductionPairSAT (1352ms), DependencyGraph (2ms), SizeChangePrinciple (1564ms), ForwardNarrowing (3ms), BackwardInstantiation (1ms), ForwardInstantiation (1ms), Propagation (1ms)].
| – Problem 3 was processed with processor SubtermCriterion (0ms).
| – Problem 4 was processed with processor SubtermCriterion (1ms).
| – Problem 5 was processed with processor SubtermCriterion (0ms).
| – Problem 6 was processed with processor SubtermCriterion (1ms).
The following open problems remain:
Open Dependency Pair Problem 2
Dependency Pairs
| f#(s(x), s(y)) | → | f#(-(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y)))) |
Rewrite Rules
| min(0, y) | → | 0 | | min(x, 0) | → | 0 |
| min(s(x), s(y)) | → | s(min(x, y)) | | max(0, y) | → | y |
| max(x, 0) | → | x | | max(s(x), s(y)) | → | s(max(x, y)) |
| twice(0) | → | 0 | | twice(s(x)) | → | s(s(twice(x))) |
| -(x, 0) | → | x | | -(s(x), s(y)) | → | -(x, y) |
| p(s(x)) | → | x | | f(s(x), s(y)) | → | f(-(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y)))) |
Original Signature
Termination of terms over the following signature is verified: f, min, twice, max, 0, s, p, -
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| f#(s(x), s(y)) | → | -#(max(s(x), s(y)), min(s(x), s(y))) | | f#(s(x), s(y)) | → | twice#(min(x, y)) |
| min#(s(x), s(y)) | → | min#(x, y) | | f#(s(x), s(y)) | → | f#(-(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y)))) |
| f#(s(x), s(y)) | → | max#(s(x), s(y)) | | f#(s(x), s(y)) | → | min#(s(x), s(y)) |
| f#(s(x), s(y)) | → | min#(x, y) | | twice#(s(x)) | → | twice#(x) |
| max#(s(x), s(y)) | → | max#(x, y) | | f#(s(x), s(y)) | → | p#(twice(min(x, y))) |
| -#(s(x), s(y)) | → | -#(x, y) |
Rewrite Rules
| min(0, y) | → | 0 | | min(x, 0) | → | 0 |
| min(s(x), s(y)) | → | s(min(x, y)) | | max(0, y) | → | y |
| max(x, 0) | → | x | | max(s(x), s(y)) | → | s(max(x, y)) |
| twice(0) | → | 0 | | twice(s(x)) | → | s(s(twice(x))) |
| -(x, 0) | → | x | | -(s(x), s(y)) | → | -(x, y) |
| p(s(x)) | → | x | | f(s(x), s(y)) | → | f(-(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y)))) |
Original Signature
Termination of terms over the following signature is verified: min, f, 0, max, twice, s, p, -
Strategy
The following SCCs where found
| min#(s(x), s(y)) → min#(x, y) |
| f#(s(x), s(y)) → f#(-(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y)))) |
| max#(s(x), s(y)) → max#(x, y) |
| -#(s(x), s(y)) → -#(x, y) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| max#(s(x), s(y)) | → | max#(x, y) |
Rewrite Rules
| min(0, y) | → | 0 | | min(x, 0) | → | 0 |
| min(s(x), s(y)) | → | s(min(x, y)) | | max(0, y) | → | y |
| max(x, 0) | → | x | | max(s(x), s(y)) | → | s(max(x, y)) |
| twice(0) | → | 0 | | twice(s(x)) | → | s(s(twice(x))) |
| -(x, 0) | → | x | | -(s(x), s(y)) | → | -(x, y) |
| p(s(x)) | → | x | | f(s(x), s(y)) | → | f(-(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y)))) |
Original Signature
Termination of terms over the following signature is verified: min, f, 0, max, twice, s, p, -
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| max#(s(x), s(y)) | → | max#(x, y) |
Problem 4: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| min#(s(x), s(y)) | → | min#(x, y) |
Rewrite Rules
| min(0, y) | → | 0 | | min(x, 0) | → | 0 |
| min(s(x), s(y)) | → | s(min(x, y)) | | max(0, y) | → | y |
| max(x, 0) | → | x | | max(s(x), s(y)) | → | s(max(x, y)) |
| twice(0) | → | 0 | | twice(s(x)) | → | s(s(twice(x))) |
| -(x, 0) | → | x | | -(s(x), s(y)) | → | -(x, y) |
| p(s(x)) | → | x | | f(s(x), s(y)) | → | f(-(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y)))) |
Original Signature
Termination of terms over the following signature is verified: min, f, 0, max, twice, s, p, -
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| min#(s(x), s(y)) | → | min#(x, y) |
Problem 5: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| -#(s(x), s(y)) | → | -#(x, y) |
Rewrite Rules
| min(0, y) | → | 0 | | min(x, 0) | → | 0 |
| min(s(x), s(y)) | → | s(min(x, y)) | | max(0, y) | → | y |
| max(x, 0) | → | x | | max(s(x), s(y)) | → | s(max(x, y)) |
| twice(0) | → | 0 | | twice(s(x)) | → | s(s(twice(x))) |
| -(x, 0) | → | x | | -(s(x), s(y)) | → | -(x, y) |
| p(s(x)) | → | x | | f(s(x), s(y)) | → | f(-(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y)))) |
Original Signature
Termination of terms over the following signature is verified: min, f, 0, max, twice, s, p, -
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| -#(s(x), s(y)) | → | -#(x, y) |
Problem 6: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
Rewrite Rules
| min(0, y) | → | 0 | | min(x, 0) | → | 0 |
| min(s(x), s(y)) | → | s(min(x, y)) | | max(0, y) | → | y |
| max(x, 0) | → | x | | max(s(x), s(y)) | → | s(max(x, y)) |
| twice(0) | → | 0 | | twice(s(x)) | → | s(s(twice(x))) |
| -(x, 0) | → | x | | -(s(x), s(y)) | → | -(x, y) |
| p(s(x)) | → | x | | f(s(x), s(y)) | → | f(-(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y)))) |
Original Signature
Termination of terms over the following signature is verified: min, f, 0, max, twice, s, p, -
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed: