TIMEOUT
The TRS could not be proven terminating. The proof attempt took 60046 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (30ms).
| – Problem 2 remains open; application of the following processors failed [SubtermCriterion (2ms), DependencyGraph (4ms), PolynomialLinearRange4iUR (360ms), DependencyGraph (3ms), PolynomialLinearRange8NegiUR (2228ms), DependencyGraph (4ms), ReductionPairSAT (timeout)].
| – Problem 3 was processed with processor SubtermCriterion (1ms).
The following open problems remain:
Open Dependency Pair Problem 2
Dependency Pairs
| g#(true, x, y, z) | → | f#(gt(x, z), x, y, s(z)) | | f#(true, x, y, z) | → | g#(gt(x, y), x, y, z) |
| g#(true, x, y, z) | → | f#(gt(x, z), x, s(y), z) |
Rewrite Rules
| f(true, x, y, z) | → | g(gt(x, y), x, y, z) | | g(true, x, y, z) | → | f(gt(x, z), x, s(y), z) |
| g(true, x, y, z) | → | f(gt(x, z), x, y, s(z)) | | 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, g, 0, s, false, true, gt
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| g#(true, x, y, z) | → | f#(gt(x, z), x, y, s(z)) | | g#(true, x, y, z) | → | gt#(x, z) |
| f#(true, x, y, z) | → | g#(gt(x, y), x, y, z) | | gt#(s(u), s(v)) | → | gt#(u, v) |
| f#(true, x, y, z) | → | gt#(x, y) | | g#(true, x, y, z) | → | f#(gt(x, z), x, s(y), z) |
Rewrite Rules
| f(true, x, y, z) | → | g(gt(x, y), x, y, z) | | g(true, x, y, z) | → | f(gt(x, z), x, s(y), z) |
| g(true, x, y, z) | → | f(gt(x, z), x, y, s(z)) | | 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, g, 0, s, true, false, gt
Strategy
The following SCCs where found
| g#(true, x, y, z) → f#(gt(x, z), x, y, s(z)) | f#(true, x, y, z) → g#(gt(x, y), x, y, z) |
| g#(true, x, y, z) → f#(gt(x, z), x, s(y), z) |
| gt#(s(u), s(v)) → gt#(u, v) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| gt#(s(u), s(v)) | → | gt#(u, v) |
Rewrite Rules
| f(true, x, y, z) | → | g(gt(x, y), x, y, z) | | g(true, x, y, z) | → | f(gt(x, z), x, s(y), z) |
| g(true, x, y, z) | → | f(gt(x, z), x, y, s(z)) | | 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, g, 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) |