MAYBE
The TRS could not be proven terminating. The proof attempt took 349 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 (0ms), DependencyGraph (2ms), PolynomialLinearRange4iUR (1ms), DependencyGraph (2ms), PolynomialLinearRange8NegiUR (0ms), DependencyGraph (5ms), PolynomialLinearRange4 (105ms), DependencyGraph (1ms), ReductionPairSAT (49ms), DependencyGraph (2ms), SizeChangePrinciple (36ms)].
The following open problems remain:
Open Dependency Pair Problem 2
Dependency Pairs
| h#(x) | → | U21#(f(x), x) | | U21#(g(x), x) | → | h#(g(x)) |
Rewrite Rules
| f(g(a)) | → | g(b) | | g(a) | → | b |
| h(x) | → | U21(f(x), x) | | U21(g(x), x) | → | h(g(x)) |
Original Signature
Termination of terms over the following signature is verified: f, g, b, a, h
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| h#(x) | → | f#(x) | | f#(g(a)) | → | g#(b) |
| h#(x) | → | U21#(f(x), x) | | U21#(g(x), x) | → | g#(x) |
| U21#(g(x), x) | → | h#(g(x)) |
Rewrite Rules
| f(g(a)) | → | g(b) | | g(a) | → | b |
| h(x) | → | U21(f(x), x) | | U21(g(x), x) | → | h(g(x)) |
Original Signature
Termination of terms over the following signature is verified: f, g, b, a, h
Strategy
Context-sensitive strategy:
μ(T) = μ(b) = μ(a) = ∅
μ(f) = μ(g) = μ(f#) = μ(U21#) = μ(h#) = μ(g#) = μ(U21) = μ(h) = {1}
The following SCCs where found
| h#(x) → U21#(f(x), x) | U21#(g(x), x) → h#(g(x)) |