YES
The TRS could be proven terminating. The proof took 602 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (14ms).
| – Problem 2 was processed with processor SubtermCriterion (1ms).
| – Problem 3 was processed with processor PolynomialOrderingProcessor (234ms).
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| log#(s(x)) | → | s#(log(half(s(x)))) | | s#(log(0)) | → | s#(0) |
| log#(s(x)) | → | half#(s(x)) | | half#(s(s(x))) | → | half#(x) |
| half#(s(s(x))) | → | s#(half(x)) | | log#(s(x)) | → | log#(half(s(x))) |
| log#(s(x)) | → | s#(x) |
Rewrite Rules
| half(0) | → | 0 | | half(s(0)) | → | 0 |
| half(s(s(x))) | → | s(half(x)) | | s(log(0)) | → | s(0) |
| log(s(x)) | → | s(log(half(s(x)))) |
Original Signature
Termination of terms over the following signature is verified: 0, s, half, log
Strategy
The following SCCs where found
| half#(s(s(x))) → half#(x) |
| log#(s(x)) → log#(half(s(x))) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| half#(s(s(x))) | → | half#(x) |
Rewrite Rules
| half(0) | → | 0 | | half(s(0)) | → | 0 |
| half(s(s(x))) | → | s(half(x)) | | s(log(0)) | → | s(0) |
| log(s(x)) | → | s(log(half(s(x)))) |
Original Signature
Termination of terms over the following signature is verified: 0, s, half, log
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| half#(s(s(x))) | → | half#(x) |
Problem 3: PolynomialOrderingProcessor
Dependency Pair Problem
Dependency Pairs
| log#(s(x)) | → | log#(half(s(x))) |
Rewrite Rules
| half(0) | → | 0 | | half(s(0)) | → | 0 |
| half(s(s(x))) | → | s(half(x)) | | s(log(0)) | → | s(0) |
| log(s(x)) | → | s(log(half(s(x)))) |
Original Signature
Termination of terms over the following signature is verified: 0, s, half, log
Strategy
Polynomial Interpretation
- 0: -1
- half(x): x - 1
- log(x): -1
- log#(x): x + 3
- s(x): x + 1
Improved Usable rules
| half(s(0)) | → | 0 | | s(log(0)) | → | s(0) |
| half(0) | → | 0 | | half(s(s(x))) | → | s(half(x)) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| log#(s(x)) | → | log#(half(s(x))) |