MAYBE
The TRS could not be proven terminating. The proof attempt took 3290 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (0ms).
| – Problem 2 was processed with processor SubtermCriterion (0ms).
| – Problem 3 was processed with processor SubtermCriterion (0ms).
| – Problem 4 remains open; application of the following processors failed [SubtermCriterion (0ms), DependencyGraph (1ms), PolynomialLinearRange4iUR (295ms), DependencyGraph (1ms), PolynomialLinearRange8NegiUR (2019ms), DependencyGraph (1ms), ReductionPairSAT (764ms), DependencyGraph (2ms), SizeChangePrinciple (41ms)].
The following open problems remain:
Open Dependency Pair Problem 4
Dependency Pairs
| f#(true, x, y) | → | f#(gt(x, y), trunc(x), s(y)) |
Rewrite Rules
| f(true, x, y) | → | f(gt(x, y), trunc(x), s(y)) | | trunc(0) | → | 0 |
| trunc(s(0)) | → | 0 | | trunc(s(s(x))) | → | s(s(trunc(x))) |
| 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, trunc, gt
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| f#(true, x, y) | → | gt#(x, y) | | f#(true, x, y) | → | f#(gt(x, y), trunc(x), s(y)) |
| trunc#(s(s(x))) | → | trunc#(x) | | gt#(s(u), s(v)) | → | gt#(u, v) |
| f#(true, x, y) | → | trunc#(x) |
Rewrite Rules
| f(true, x, y) | → | f(gt(x, y), trunc(x), s(y)) | | trunc(0) | → | 0 |
| trunc(s(0)) | → | 0 | | trunc(s(s(x))) | → | s(s(trunc(x))) |
| 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, trunc
Strategy
The following SCCs where found
| f#(true, x, y) → f#(gt(x, y), trunc(x), s(y)) |
| trunc#(s(s(x))) → trunc#(x) |
| gt#(s(u), s(v)) → gt#(u, v) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| trunc#(s(s(x))) | → | trunc#(x) |
Rewrite Rules
| f(true, x, y) | → | f(gt(x, y), trunc(x), s(y)) | | trunc(0) | → | 0 |
| trunc(s(0)) | → | 0 | | trunc(s(s(x))) | → | s(s(trunc(x))) |
| 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, trunc
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| trunc#(s(s(x))) | → | trunc#(x) |
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), trunc(x), s(y)) | | trunc(0) | → | 0 |
| trunc(s(0)) | → | 0 | | trunc(s(s(x))) | → | s(s(trunc(x))) |
| 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, trunc
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| gt#(s(u), s(v)) | → | gt#(u, v) |