YES
The TRS could be proven terminating. The proof took 598 ms.
The following DP Processors were used
Problem 1 was processed with processor PolynomialLinearRange4iUR (233ms).
| – Problem 2 was processed with processor PolynomialLinearRange4iUR (118ms).
Problem 1: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| f#(s(s(x))) | → | f#(s(x)) | | f#(s(s(x))) | → | f#(f(s(x))) |
Rewrite Rules
| f(0) | → | s(0) | | f(s(0)) | → | s(0) |
| f(s(s(x))) | → | f(f(s(x))) |
Original Signature
Termination of terms over the following signature is verified: f, 0, s
Strategy
Polynomial Interpretation
- 0: 0
- f(x): 2
- f#(x): x + 1
- s(x): x + 1
Improved Usable rules
| f(0) | → | s(0) | | f(s(s(x))) | → | f(f(s(x))) |
| f(s(0)) | → | s(0) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
Problem 2: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| f#(s(s(x))) | → | f#(f(s(x))) |
Rewrite Rules
| f(0) | → | s(0) | | f(s(0)) | → | s(0) |
| f(s(s(x))) | → | f(f(s(x))) |
Original Signature
Termination of terms over the following signature is verified: f, 0, s
Strategy
Polynomial Interpretation
- 0: 0
- f(x): 1
- f#(x): x + 1
- s(x): x + 1
Improved Usable rules
| f(0) | → | s(0) | | f(s(s(x))) | → | f(f(s(x))) |
| f(s(0)) | → | s(0) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| f#(s(s(x))) | → | f#(f(s(x))) |