YES
The TRS could be proven terminating. The proof took 131 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (3ms).
| – Problem 2 was processed with processor PolynomialLinearRange4 (79ms).
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| f#(s(0)) | → | f#(p(s(0))) | | T(f(s(0))) | → | f#(s(0)) |
| f#(s(0)) | → | p#(s(0)) |
Rewrite Rules
| f(0) | → | cons(0, f(s(0))) | | f(s(0)) | → | f(p(s(0))) |
| p(s(0)) | → | 0 |
Original Signature
Termination of terms over the following signature is verified: f, 0, s, p, cons
Strategy
Context-sensitive strategy:
μ(T) = μ(0) = ∅
μ(f) = μ(f#) = μ(p#) = μ(s) = μ(p) = μ(cons) = {1}
The following SCCs where found
Problem 2: PolynomialLinearRange4
Dependency Pair Problem
Dependency Pairs
Rewrite Rules
| f(0) | → | cons(0, f(s(0))) | | f(s(0)) | → | f(p(s(0))) |
| p(s(0)) | → | 0 |
Original Signature
Termination of terms over the following signature is verified: f, 0, s, p, cons
Strategy
Context-sensitive strategy:
μ(T) = μ(0) = ∅
μ(f) = μ(f#) = μ(p#) = μ(s) = μ(p) = μ(cons) = {1}
Polynomial Interpretation
- 0: 1
- cons(x,y): 0
- f(x): 0
- f#(x): x
- p(x): 2
- s(x): 2x + 1
Standard Usable rules
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed: