YES
The TRS could be proven terminating. The proof took 224 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (4ms).
| – Problem 2 was processed with processor PolynomialOrderingProcessor (66ms).
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| f#(s(0)) | → | f#(p(s(0))) | | activate#(n__f(X)) | → | f#(X) |
| f#(s(0)) | → | p#(s(0)) |
Rewrite Rules
| f(0) | → | cons(0, n__f(s(0))) | | f(s(0)) | → | f(p(s(0))) |
| p(s(X)) | → | X | | f(X) | → | n__f(X) |
| activate(n__f(X)) | → | f(X) | | activate(X) | → | X |
Original Signature
Termination of terms over the following signature is verified: f, activate, 0, s, p, n__f, cons
Strategy
The following SCCs where found
Problem 2: PolynomialOrderingProcessor
Dependency Pair Problem
Dependency Pairs
Rewrite Rules
| f(0) | → | cons(0, n__f(s(0))) | | f(s(0)) | → | f(p(s(0))) |
| p(s(X)) | → | X | | f(X) | → | n__f(X) |
| activate(n__f(X)) | → | f(X) | | activate(X) | → | X |
Original Signature
Termination of terms over the following signature is verified: f, activate, 0, s, p, n__f, cons
Strategy
Polynomial Interpretation
- 0: -1
- activate(x): -2
- cons(x,y): -2
- f(x): -2
- f#(x): x - 1
- n__f(x): -2
- p(x): x - 2
- s(x): 4x + 2
Improved Usable rules
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed: