TIMEOUT
The TRS could not be proven terminating. The proof attempt took 60029 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (67ms).
| – Problem 2 was processed with processor SubtermCriterion (2ms).
| – 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).
| – Problem 7 remains open; application of the following processors failed [SubtermCriterion (0ms), DependencyGraph (1ms), PolynomialLinearRange4iUR (1793ms), DependencyGraph (1ms), PolynomialLinearRange8NegiUR (29996ms), DependencyGraph (timeout), ReductionPairSAT (1218ms), DependencyGraph (2ms), SizeChangePrinciple (950ms), ForwardNarrowing (0ms), BackwardInstantiation (1ms), ForwardInstantiation (1ms), Propagation (1ms)].
The following open problems remain:
Open Dependency Pair Problem 7
Dependency Pairs
| f#(s(x)) | → | f#(-(max(*(s(x), s(x)), +(s(x), s(s(s(0))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0)))))))) |
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)) |
| +(0, y) | → | y | | +(s(x), y) | → | s(+(x, y)) |
| -(x, 0) | → | x | | -(s(x), s(y)) | → | -(x, y) |
| *(x, 0) | → | 0 | | *(x, s(y)) | → | +(x, *(x, y)) |
| f(s(x)) | → | f(-(max(*(s(x), s(x)), +(s(x), s(s(s(0))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0)))))))) |
Original Signature
Termination of terms over the following signature is verified: f, min, max, 0, s, *, +, -
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| f#(s(x)) | → | -#(max(*(s(x), s(x)), +(s(x), s(s(s(0))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0))))))) | | *#(x, s(y)) | → | *#(x, y) |
| f#(s(x)) | → | +#(s(x), s(s(s(s(0))))) | | f#(s(x)) | → | max#(s(*(s(x), s(x))), +(s(x), s(s(s(s(0)))))) |
| max#(s(x), s(y)) | → | max#(x, y) | | f#(s(x)) | → | +#(s(x), s(s(s(0)))) |
| +#(s(x), y) | → | +#(x, y) | | -#(s(x), s(y)) | → | -#(x, y) |
| *#(x, s(y)) | → | +#(x, *(x, y)) | | f#(s(x)) | → | f#(-(max(*(s(x), s(x)), +(s(x), s(s(s(0))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0)))))))) |
| min#(s(x), s(y)) | → | min#(x, y) | | f#(s(x)) | → | *#(s(x), s(x)) |
| f#(s(x)) | → | max#(*(s(x), s(x)), +(s(x), s(s(s(0))))) |
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)) |
| +(0, y) | → | y | | +(s(x), y) | → | s(+(x, y)) |
| -(x, 0) | → | x | | -(s(x), s(y)) | → | -(x, y) |
| *(x, 0) | → | 0 | | *(x, s(y)) | → | +(x, *(x, y)) |
| f(s(x)) | → | f(-(max(*(s(x), s(x)), +(s(x), s(s(s(0))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0)))))))) |
Original Signature
Termination of terms over the following signature is verified: min, f, 0, max, s, *, +, -
Strategy
The following SCCs where found
| f#(s(x)) → f#(-(max(*(s(x), s(x)), +(s(x), s(s(s(0))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0)))))))) |
| min#(s(x), s(y)) → min#(x, y) |
| max#(s(x), s(y)) → max#(x, y) |
| -#(s(x), s(y)) → -#(x, y) |
Problem 2: 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)) |
| +(0, y) | → | y | | +(s(x), y) | → | s(+(x, y)) |
| -(x, 0) | → | x | | -(s(x), s(y)) | → | -(x, y) |
| *(x, 0) | → | 0 | | *(x, s(y)) | → | +(x, *(x, y)) |
| f(s(x)) | → | f(-(max(*(s(x), s(x)), +(s(x), s(s(s(0))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0)))))))) |
Original Signature
Termination of terms over the following signature is verified: min, f, 0, max, s, *, +, -
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
Problem 3: 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)) |
| +(0, y) | → | y | | +(s(x), y) | → | s(+(x, y)) |
| -(x, 0) | → | x | | -(s(x), s(y)) | → | -(x, y) |
| *(x, 0) | → | 0 | | *(x, s(y)) | → | +(x, *(x, y)) |
| f(s(x)) | → | f(-(max(*(s(x), s(x)), +(s(x), s(s(s(0))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0)))))))) |
Original Signature
Termination of terms over the following signature is verified: min, f, 0, max, s, *, +, -
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
Problem 4: 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)) |
| +(0, y) | → | y | | +(s(x), y) | → | s(+(x, y)) |
| -(x, 0) | → | x | | -(s(x), s(y)) | → | -(x, y) |
| *(x, 0) | → | 0 | | *(x, s(y)) | → | +(x, *(x, y)) |
| f(s(x)) | → | f(-(max(*(s(x), s(x)), +(s(x), s(s(s(0))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0)))))))) |
Original Signature
Termination of terms over the following signature is verified: min, f, 0, max, s, *, +, -
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| -#(s(x), s(y)) | → | -#(x, y) |
Problem 5: 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)) |
| +(0, y) | → | y | | +(s(x), y) | → | s(+(x, y)) |
| -(x, 0) | → | x | | -(s(x), s(y)) | → | -(x, y) |
| *(x, 0) | → | 0 | | *(x, s(y)) | → | +(x, *(x, y)) |
| f(s(x)) | → | f(-(max(*(s(x), s(x)), +(s(x), s(s(s(0))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0)))))))) |
Original Signature
Termination of terms over the following signature is verified: min, f, 0, max, s, *, +, -
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| max#(s(x), s(y)) | → | max#(x, y) |
Problem 6: 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)) |
| +(0, y) | → | y | | +(s(x), y) | → | s(+(x, y)) |
| -(x, 0) | → | x | | -(s(x), s(y)) | → | -(x, y) |
| *(x, 0) | → | 0 | | *(x, s(y)) | → | +(x, *(x, y)) |
| f(s(x)) | → | f(-(max(*(s(x), s(x)), +(s(x), s(s(s(0))))), max(s(*(s(x), s(x))), +(s(x), s(s(s(s(0)))))))) |
Original Signature
Termination of terms over the following signature is verified: min, f, 0, max, s, *, +, -
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| min#(s(x), s(y)) | → | min#(x, y) |