YES
The TRS could be proven terminating. The proof took 60000 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (5ms).
| – Problem 2 was processed with processor PolynomialOrderingProcessor (382ms).
| | – Problem 3 was processed with processor PolynomialOrderingProcessor (213ms).
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| f#(a, f(x, a)) | → | f#(a, a) | | f#(a, f(x, a)) | → | f#(f(a, a), a) |
| f#(a, f(x, a)) | → | f#(a, f(x, f(f(a, a), a))) | | f#(a, f(x, a)) | → | f#(x, f(f(a, a), a)) |
Rewrite Rules
| f(a, f(x, a)) | → | f(a, f(x, f(f(a, a), a))) |
Original Signature
Termination of terms over the following signature is verified: f, a
Strategy
The following SCCs where found
| f#(a, f(x, a)) → f#(a, f(x, f(f(a, a), a))) | f#(a, f(x, a)) → f#(x, f(f(a, a), a)) |
Problem 2: PolynomialOrderingProcessor
Dependency Pair Problem
Dependency Pairs
| f#(a, f(x, a)) | → | f#(a, f(x, f(f(a, a), a))) | | f#(a, f(x, a)) | → | f#(x, f(f(a, a), a)) |
Rewrite Rules
| f(a, f(x, a)) | → | f(a, f(x, f(f(a, a), a))) |
Original Signature
Termination of terms over the following signature is verified: f, a
Strategy
Polynomial Interpretation
- a: 4
- f(x,y): y - 2
- f#(x,y): y - 1
Improved Usable rules
| f(a, f(x, a)) | → | f(a, f(x, f(f(a, a), a))) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| f#(a, f(x, a)) | → | f#(a, f(x, f(f(a, a), a))) |
Problem 3: PolynomialOrderingProcessor
Dependency Pair Problem
Dependency Pairs
| f#(a, f(x, a)) | → | f#(x, f(f(a, a), a)) |
Rewrite Rules
| f(a, f(x, a)) | → | f(a, f(x, f(f(a, a), a))) |
Original Signature
Termination of terms over the following signature is verified: f, a
Strategy
Polynomial Interpretation
- a: 3
- f(x,y): x - 1
- f#(x,y): y + x + 2
Improved Usable rules
| f(a, f(x, a)) | → | f(a, f(x, f(f(a, a), a))) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| f#(a, f(x, a)) | → | f#(x, f(f(a, a), a)) |