YES
The TRS could be proven terminating. The proof took 556 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (18ms).
| – Problem 2 was processed with processor SubtermCriterion (1ms).
| – Problem 3 was processed with processor PolynomialLinearRange4iUR (362ms).
| | – Problem 4 was processed with processor DependencyGraph (1ms).
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| g#(s(x)) | → | -#(s(x), f(g(x))) | | f#(s(x)) | → | g#(f(x)) |
| g#(s(x)) | → | f#(g(x)) | | f#(s(x)) | → | -#(s(x), g(f(x))) |
| -#(s(x), s(y)) | → | -#(x, y) | | f#(s(x)) | → | f#(x) |
| g#(s(x)) | → | g#(x) |
Rewrite Rules
| -(x, 0) | → | x | | -(0, s(y)) | → | 0 |
| -(s(x), s(y)) | → | -(x, y) | | f(0) | → | 0 |
| f(s(x)) | → | -(s(x), g(f(x))) | | g(0) | → | s(0) |
| g(s(x)) | → | -(s(x), f(g(x))) |
Original Signature
Termination of terms over the following signature is verified: f, g, 0, s, -
Strategy
The following SCCs where found
| f#(s(x)) → g#(f(x)) | g#(s(x)) → f#(g(x)) |
| f#(s(x)) → f#(x) | g#(s(x)) → g#(x) |
| -#(s(x), s(y)) → -#(x, y) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| -#(s(x), s(y)) | → | -#(x, y) |
Rewrite Rules
| -(x, 0) | → | x | | -(0, s(y)) | → | 0 |
| -(s(x), s(y)) | → | -(x, y) | | f(0) | → | 0 |
| f(s(x)) | → | -(s(x), g(f(x))) | | g(0) | → | s(0) |
| g(s(x)) | → | -(s(x), f(g(x))) |
Original Signature
Termination of terms over the following signature is verified: f, g, 0, s, -
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| -#(s(x), s(y)) | → | -#(x, y) |
Problem 3: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| f#(s(x)) | → | g#(f(x)) | | g#(s(x)) | → | f#(g(x)) |
| f#(s(x)) | → | f#(x) | | g#(s(x)) | → | g#(x) |
Rewrite Rules
| -(x, 0) | → | x | | -(0, s(y)) | → | 0 |
| -(s(x), s(y)) | → | -(x, y) | | f(0) | → | 0 |
| f(s(x)) | → | -(s(x), g(f(x))) | | g(0) | → | s(0) |
| g(s(x)) | → | -(s(x), f(g(x))) |
Original Signature
Termination of terms over the following signature is verified: f, g, 0, s, -
Strategy
Polynomial Interpretation
- -(x,y): x
- 0: 0
- f(x): 2x
- f#(x): 2x
- g(x): x + 1
- g#(x): x + 2
- s(x): 2x + 1
Improved Usable rules
| -(s(x), s(y)) | → | -(x, y) | | g(0) | → | s(0) |
| f(s(x)) | → | -(s(x), g(f(x))) | | -(x, 0) | → | x |
| -(0, s(y)) | → | 0 | | g(s(x)) | → | -(s(x), f(g(x))) |
| f(0) | → | 0 |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| g#(s(x)) | → | f#(g(x)) | | g#(s(x)) | → | g#(x) |
| f#(s(x)) | → | f#(x) |
Problem 4: DependencyGraph
Dependency Pair Problem
Dependency Pairs
Rewrite Rules
| -(x, 0) | → | x | | -(0, s(y)) | → | 0 |
| -(s(x), s(y)) | → | -(x, y) | | f(0) | → | 0 |
| f(s(x)) | → | -(s(x), g(f(x))) | | g(0) | → | s(0) |
| g(s(x)) | → | -(s(x), f(g(x))) |
Original Signature
Termination of terms over the following signature is verified: f, g, 0, s, -
Strategy
There are no SCCs!