TIMEOUT
The TRS could not be proven terminating. The proof attempt took 60001 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (38ms).
| – Problem 2 was processed with processor SubtermCriterion (1ms).
| – Problem 3 was processed with processor SubtermCriterion (1ms).
| – Problem 4 was processed with processor SubtermCriterion (1ms).
| – Problem 5 remains open; application of the following processors failed [SubtermCriterion (1ms), DependencyGraph (1ms), PolynomialLinearRange4iUR (764ms), DependencyGraph (2ms), PolynomialLinearRange8NegiUR (12257ms), DependencyGraph (2ms), ReductionPairSAT (879ms), DependencyGraph (1ms), SizeChangePrinciple (260ms), ForwardNarrowing (1ms), BackwardInstantiation (1ms), ForwardInstantiation (1ms), Propagation (3ms)].
The following open problems remain:
Open Dependency Pair Problem 5
Dependency Pairs
| gcd#(s(x), s(y)) | → | gcd#(-(s(max(x, y)), s(min(x, y))), s(min(x, y))) |
Rewrite Rules
| min(x, 0) | → | 0 | | min(0, y) | → | 0 |
| min(s(x), s(y)) | → | s(min(x, y)) | | max(x, 0) | → | x |
| max(0, y) | → | y | | max(s(x), s(y)) | → | s(max(x, y)) |
| -(x, 0) | → | x | | -(s(x), s(y)) | → | -(x, y) |
| gcd(s(x), s(y)) | → | gcd(-(s(max(x, y)), s(min(x, y))), s(min(x, y))) | | gcd(s(x), 0) | → | s(x) |
| gcd(0, s(y)) | → | s(y) |
Original Signature
Termination of terms over the following signature is verified: min, max, 0, s, gcd, -
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| gcd#(s(x), s(y)) | → | gcd#(-(s(max(x, y)), s(min(x, y))), s(min(x, y))) | | min#(s(x), s(y)) | → | min#(x, y) |
| gcd#(s(x), s(y)) | → | max#(x, y) | | gcd#(s(x), s(y)) | → | min#(x, y) |
| max#(s(x), s(y)) | → | max#(x, y) | | -#(s(x), s(y)) | → | -#(x, y) |
| gcd#(s(x), s(y)) | → | -#(s(max(x, y)), s(min(x, y))) |
Rewrite Rules
| min(x, 0) | → | 0 | | min(0, y) | → | 0 |
| min(s(x), s(y)) | → | s(min(x, y)) | | max(x, 0) | → | x |
| max(0, y) | → | y | | max(s(x), s(y)) | → | s(max(x, y)) |
| -(x, 0) | → | x | | -(s(x), s(y)) | → | -(x, y) |
| gcd(s(x), s(y)) | → | gcd(-(s(max(x, y)), s(min(x, y))), s(min(x, y))) | | gcd(s(x), 0) | → | s(x) |
| gcd(0, s(y)) | → | s(y) |
Original Signature
Termination of terms over the following signature is verified: min, 0, max, s, -, gcd
Strategy
The following SCCs where found
| gcd#(s(x), s(y)) → gcd#(-(s(max(x, y)), s(min(x, y))), s(min(x, y))) |
| 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
| -#(s(x), s(y)) | → | -#(x, y) |
Rewrite Rules
| min(x, 0) | → | 0 | | min(0, y) | → | 0 |
| min(s(x), s(y)) | → | s(min(x, y)) | | max(x, 0) | → | x |
| max(0, y) | → | y | | max(s(x), s(y)) | → | s(max(x, y)) |
| -(x, 0) | → | x | | -(s(x), s(y)) | → | -(x, y) |
| gcd(s(x), s(y)) | → | gcd(-(s(max(x, y)), s(min(x, y))), s(min(x, y))) | | gcd(s(x), 0) | → | s(x) |
| gcd(0, s(y)) | → | s(y) |
Original Signature
Termination of terms over the following signature is verified: min, 0, max, s, -, gcd
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| -#(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(x, 0) | → | 0 | | min(0, y) | → | 0 |
| min(s(x), s(y)) | → | s(min(x, y)) | | max(x, 0) | → | x |
| max(0, y) | → | y | | max(s(x), s(y)) | → | s(max(x, y)) |
| -(x, 0) | → | x | | -(s(x), s(y)) | → | -(x, y) |
| gcd(s(x), s(y)) | → | gcd(-(s(max(x, y)), s(min(x, y))), s(min(x, y))) | | gcd(s(x), 0) | → | s(x) |
| gcd(0, s(y)) | → | s(y) |
Original Signature
Termination of terms over the following signature is verified: min, 0, max, s, -, gcd
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(x, 0) | → | 0 | | min(0, y) | → | 0 |
| min(s(x), s(y)) | → | s(min(x, y)) | | max(x, 0) | → | x |
| max(0, y) | → | y | | max(s(x), s(y)) | → | s(max(x, y)) |
| -(x, 0) | → | x | | -(s(x), s(y)) | → | -(x, y) |
| gcd(s(x), s(y)) | → | gcd(-(s(max(x, y)), s(min(x, y))), s(min(x, y))) | | gcd(s(x), 0) | → | s(x) |
| gcd(0, s(y)) | → | s(y) |
Original Signature
Termination of terms over the following signature is verified: min, 0, max, s, -, gcd
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| min#(s(x), s(y)) | → | min#(x, y) |