YES
The TRS could be proven terminating. The proof took 129 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (5ms).
| – Problem 2 was processed with processor PolynomialLinearRange4iUR (101ms).
| – Problem 3 was processed with processor SubtermCriterion (0ms).
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| h#(g(x)) | → | h#(a) | | f#(a, x) | → | f#(g(x), x) |
| f#(a, x) | → | g#(x) | | g#(h(x)) | → | g#(x) |
Rewrite Rules
| f(a, x) | → | f(g(x), x) | | h(g(x)) | → | h(a) |
| g(h(x)) | → | g(x) | | h(h(x)) | → | x |
Original Signature
Termination of terms over the following signature is verified: f, g, a, h
Strategy
The following SCCs where found
Problem 2: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
Rewrite Rules
| f(a, x) | → | f(g(x), x) | | h(g(x)) | → | h(a) |
| g(h(x)) | → | g(x) | | h(h(x)) | → | x |
Original Signature
Termination of terms over the following signature is verified: f, g, a, h
Strategy
Polynomial Interpretation
- a: 2
- f(x,y): 0
- f#(x,y): x
- g(x): 0
- h(x): 3
Improved Usable rules
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
Rewrite Rules
| f(a, x) | → | f(g(x), x) | | h(g(x)) | → | h(a) |
| g(h(x)) | → | g(x) | | h(h(x)) | → | x |
Original Signature
Termination of terms over the following signature is verified: f, g, a, h
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed: