YES
The TRS could be proven terminating. The proof took 622 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (17ms).
| – Problem 2 was processed with processor SubtermCriterion (1ms).
| – Problem 3 was processed with processor PolynomialLinearRange4iUR (268ms).
| | – Problem 4 was processed with processor PolynomialLinearRange4iUR (144ms).
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| g#(g(x)) | → | h#(g(x)) | | g#(g(x)) | → | g#(x) |
| h#(h(x)) | → | h#(f(h(x), x)) | | h#(h(x)) | → | h#(x) |
| g#(h(g(x))) | → | g#(x) | | g#(g(x)) | → | g#(h(g(x))) |
Rewrite Rules
| g(h(g(x))) | → | g(x) | | g(g(x)) | → | g(h(g(x))) |
| h(h(x)) | → | h(f(h(x), x)) |
Original Signature
Termination of terms over the following signature is verified: f, g, h
Strategy
The following SCCs where found
| g#(g(x)) → g#(x) | g#(h(g(x))) → g#(x) |
| g#(g(x)) → g#(h(g(x))) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
Rewrite Rules
| g(h(g(x))) | → | g(x) | | g(g(x)) | → | g(h(g(x))) |
| h(h(x)) | → | h(f(h(x), x)) |
Original Signature
Termination of terms over the following signature is verified: f, g, h
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
Problem 3: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| g#(g(x)) | → | g#(x) | | g#(h(g(x))) | → | g#(x) |
| g#(g(x)) | → | g#(h(g(x))) |
Rewrite Rules
| g(h(g(x))) | → | g(x) | | g(g(x)) | → | g(h(g(x))) |
| h(h(x)) | → | h(f(h(x), x)) |
Original Signature
Termination of terms over the following signature is verified: f, g, h
Strategy
Polynomial Interpretation
- f(x,y): y
- g(x): x + 1
- g#(x): x
- h(x): x
Improved Usable rules
| h(h(x)) | → | h(f(h(x), x)) | | g(h(g(x))) | → | g(x) |
| g(g(x)) | → | g(h(g(x))) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| g#(g(x)) | → | g#(x) | | g#(h(g(x))) | → | g#(x) |
Problem 4: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
Rewrite Rules
| g(h(g(x))) | → | g(x) | | g(g(x)) | → | g(h(g(x))) |
| h(h(x)) | → | h(f(h(x), x)) |
Original Signature
Termination of terms over the following signature is verified: f, g, h
Strategy
Polynomial Interpretation
- f(x,y): 0
- g(x): 1
- g#(x): 2x + 1
- h(x): 0
Improved Usable rules
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed: