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 (11ms).
| – Problem 2 remains open; application of the following processors failed [SubtermCriterion (1ms), DependencyGraph (1ms), PolynomialLinearRange4iUR (152ms), DependencyGraph (2ms), PolynomialLinearRange8NegiUR (898ms), DependencyGraph (1ms), ReductionPairSAT (262ms), DependencyGraph (1ms), SizeChangePrinciple (9ms), ForwardNarrowing (1ms), BackwardInstantiation (1ms), ForwardInstantiation (0ms), Propagation (1ms)].
| – Problem 3 was processed with processor SubtermCriterion (0ms).
The following open problems remain:
Open Dependency Pair Problem 2
Dependency Pairs
| f#(g(x), s(0)) | → | f#(g(x), g(x)) |
Rewrite Rules
| f(g(x), s(0)) | → | f(g(x), g(x)) | | g(s(x)) | → | s(g(x)) |
| g(0) | → | 0 |
Original Signature
Termination of terms over the following signature is verified: f, g, 0, s
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| f#(g(x), s(0)) | → | g#(x) | | f#(g(x), s(0)) | → | f#(g(x), g(x)) |
| g#(s(x)) | → | g#(x) |
Rewrite Rules
| f(g(x), s(0)) | → | f(g(x), g(x)) | | g(s(x)) | → | s(g(x)) |
| g(0) | → | 0 |
Original Signature
Termination of terms over the following signature is verified: f, g, 0, s
Strategy
The following SCCs where found
| f#(g(x), s(0)) → f#(g(x), g(x)) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
Rewrite Rules
| f(g(x), s(0)) | → | f(g(x), g(x)) | | g(s(x)) | → | s(g(x)) |
| g(0) | → | 0 |
Original Signature
Termination of terms over the following signature is verified: f, g, 0, s
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed: