MAYBE
The TRS could not be proven terminating. The proof attempt took 729 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 (147ms), DependencyGraph (1ms), PolynomialLinearRange8NegiUR (240ms), DependencyGraph (1ms), ReductionPairSAT (191ms), DependencyGraph (2ms), SizeChangePrinciple (19ms)].
| – Problem 3 was processed with processor SubtermCriterion (0ms).
The following open problems remain:
Open Dependency Pair Problem 2
Dependency Pairs
| f#(X, Y, g(X, Y)) | → | h#(0, g(X, Y)) | | h#(X, Z) | → | f#(X, s(X), Z) |
Rewrite Rules
| h(X, Z) | → | f(X, s(X), Z) | | f(X, Y, g(X, Y)) | → | h(0, g(X, Y)) |
| g(0, Y) | → | 0 | | g(X, s(Y)) | → | g(X, Y) |
Original Signature
Termination of terms over the following signature is verified: f, g, 0, s, h
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| f#(X, Y, g(X, Y)) | → | g#(X, Y) | | f#(X, Y, g(X, Y)) | → | h#(0, g(X, Y)) |
| h#(X, Z) | → | f#(X, s(X), Z) | | g#(X, s(Y)) | → | g#(X, Y) |
Rewrite Rules
| h(X, Z) | → | f(X, s(X), Z) | | f(X, Y, g(X, Y)) | → | h(0, g(X, Y)) |
| g(0, Y) | → | 0 | | g(X, s(Y)) | → | g(X, Y) |
Original Signature
Termination of terms over the following signature is verified: f, g, 0, s, h
Strategy
The following SCCs where found
| f#(X, Y, g(X, Y)) → h#(0, g(X, Y)) | h#(X, Z) → f#(X, s(X), Z) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
Rewrite Rules
| h(X, Z) | → | f(X, s(X), Z) | | f(X, Y, g(X, Y)) | → | h(0, g(X, Y)) |
| g(0, Y) | → | 0 | | g(X, s(Y)) | → | g(X, Y) |
Original Signature
Termination of terms over the following signature is verified: f, g, 0, s, h
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed: