MAYBE
The TRS could not be proven terminating. The proof attempt took 1550 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 (2ms), PolynomialLinearRange4iUR (205ms), DependencyGraph (2ms), PolynomialLinearRange8NegiUR (561ms), DependencyGraph (1ms), ReductionPairSAT (659ms), DependencyGraph (1ms), SizeChangePrinciple (13ms)].
| – Problem 3 was processed with processor SubtermCriterion (0ms).
The following open problems remain:
Open Dependency Pair Problem 2
Dependency Pairs
| f#(true, x, y) | → | f#(gt(x, y), s(x), s(s(y))) |
Rewrite Rules
| f(true, x, y) | → | f(gt(x, y), s(x), s(s(y))) | | gt(0, v) | → | false |
| gt(s(u), 0) | → | true | | gt(s(u), s(v)) | → | gt(u, v) |
Original Signature
Termination of terms over the following signature is verified: f, 0, s, false, true, gt
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| f#(true, x, y) | → | gt#(x, y) | | gt#(s(u), s(v)) | → | gt#(u, v) |
| f#(true, x, y) | → | f#(gt(x, y), s(x), s(s(y))) |
Rewrite Rules
| f(true, x, y) | → | f(gt(x, y), s(x), s(s(y))) | | gt(0, v) | → | false |
| gt(s(u), 0) | → | true | | gt(s(u), s(v)) | → | gt(u, v) |
Original Signature
Termination of terms over the following signature is verified: f, 0, s, true, false, gt
Strategy
The following SCCs where found
| gt#(s(u), s(v)) → gt#(u, v) |
| f#(true, x, y) → f#(gt(x, y), s(x), s(s(y))) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| gt#(s(u), s(v)) | → | gt#(u, v) |
Rewrite Rules
| f(true, x, y) | → | f(gt(x, y), s(x), s(s(y))) | | gt(0, v) | → | false |
| gt(s(u), 0) | → | true | | gt(s(u), s(v)) | → | gt(u, v) |
Original Signature
Termination of terms over the following signature is verified: f, 0, s, true, false, gt
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| gt#(s(u), s(v)) | → | gt#(u, v) |