YES
The TRS could be proven terminating. The proof took 136 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (8ms).
| – Problem 2 was processed with processor PolynomialLinearRange4iUR (92ms).
| – Problem 3 was processed with processor PolynomialLinearRange4iUR (17ms).
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| f#(h(x)) | → | f#(i(x)) | | g#(i(x)) | → | g#(h(x)) |
| g#(i(x)) | → | h#(x) | | f#(h(x)) | → | i#(x) |
Rewrite Rules
| f(h(x)) | → | f(i(x)) | | g(i(x)) | → | g(h(x)) |
| h(a) | → | b | | i(a) | → | b |
Original Signature
Termination of terms over the following signature is verified: f, g, b, a, h, i
Strategy
The following SCCs where found
Problem 2: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
Rewrite Rules
| f(h(x)) | → | f(i(x)) | | g(i(x)) | → | g(h(x)) |
| h(a) | → | b | | i(a) | → | b |
Original Signature
Termination of terms over the following signature is verified: f, g, b, a, h, i
Strategy
Polynomial Interpretation
- a: 1
- b: 0
- f(x): 0
- g(x): 0
- g#(x): x + 1
- h(x): 0
- i(x): 1
Improved Usable rules
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
Problem 3: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
Rewrite Rules
| f(h(x)) | → | f(i(x)) | | g(i(x)) | → | g(h(x)) |
| h(a) | → | b | | i(a) | → | b |
Original Signature
Termination of terms over the following signature is verified: f, g, b, a, h, i
Strategy
Polynomial Interpretation
- a: 1
- b: 0
- f(x): 0
- f#(x): 2x
- g(x): 0
- h(x): 1
- i(x): 0
Improved Usable rules
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed: