MAYBE
The TRS could not be proven terminating. The proof attempt took 1215 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (0ms).
| – Problem 2 remains open; application of the following processors failed [SubtermCriterion (1ms), DependencyGraph (3ms), PolynomialLinearRange4iUR (155ms), DependencyGraph (2ms), PolynomialLinearRange8NegiUR (447ms), DependencyGraph (0ms), ReductionPairSAT (445ms), DependencyGraph (1ms), SizeChangePrinciple (14ms)].
| – Problem 3 was processed with processor SubtermCriterion (0ms).
The following open problems remain:
Open Dependency Pair Problem 2
Dependency Pairs
| f#(t, x, y) | → | f#(g(x, y), x, s(y)) |
Rewrite Rules
| f(t, x, y) | → | f(g(x, y), x, s(y)) | | g(s(x), 0) | → | t |
| g(s(x), s(y)) | → | g(x, y) |
Original Signature
Termination of terms over the following signature is verified: f, g, 0, t, s
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| f#(t, x, y) | → | g#(x, y) | | g#(s(x), s(y)) | → | g#(x, y) |
| f#(t, x, y) | → | f#(g(x, y), x, s(y)) |
Rewrite Rules
| f(t, x, y) | → | f(g(x, y), x, s(y)) | | g(s(x), 0) | → | t |
| g(s(x), s(y)) | → | g(x, y) |
Original Signature
Termination of terms over the following signature is verified: f, g, t, 0, s
Strategy
The following SCCs where found
| g#(s(x), s(y)) → g#(x, y) |
| f#(t, x, y) → f#(g(x, y), x, s(y)) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| g#(s(x), s(y)) | → | g#(x, y) |
Rewrite Rules
| f(t, x, y) | → | f(g(x, y), x, s(y)) | | g(s(x), 0) | → | t |
| g(s(x), s(y)) | → | g(x, y) |
Original Signature
Termination of terms over the following signature is verified: f, g, t, 0, s
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| g#(s(x), s(y)) | → | g#(x, y) |