YES
The TRS could be proven terminating. The proof took 259 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (7ms).
| – Problem 2 was processed with processor SubtermCriterion (0ms).
| – Problem 3 was processed with processor PolynomialLinearRange4iUR (175ms).
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| quot#(s(x), s(y)) | → | minus#(x, y) | | minus#(x, s(y)) | → | minus#(x, y) |
| quot#(s(x), s(y)) | → | quot#(minus(x, y), s(y)) | | minus#(x, s(y)) | → | pred#(minus(x, y)) |
Rewrite Rules
| pred(s(x)) | → | x | | minus(x, 0) | → | x |
| minus(x, s(y)) | → | pred(minus(x, y)) | | quot(0, s(y)) | → | 0 |
| quot(s(x), s(y)) | → | s(quot(minus(x, y), s(y))) |
Original Signature
Termination of terms over the following signature is verified: minus, 0, s, pred, quot
Strategy
The following SCCs where found
| quot#(s(x), s(y)) → quot#(minus(x, y), s(y)) |
| minus#(x, s(y)) → minus#(x, y) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| minus#(x, s(y)) | → | minus#(x, y) |
Rewrite Rules
| pred(s(x)) | → | x | | minus(x, 0) | → | x |
| minus(x, s(y)) | → | pred(minus(x, y)) | | quot(0, s(y)) | → | 0 |
| quot(s(x), s(y)) | → | s(quot(minus(x, y), s(y))) |
Original Signature
Termination of terms over the following signature is verified: minus, 0, s, pred, quot
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| minus#(x, s(y)) | → | minus#(x, y) |
Problem 3: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| quot#(s(x), s(y)) | → | quot#(minus(x, y), s(y)) |
Rewrite Rules
| pred(s(x)) | → | x | | minus(x, 0) | → | x |
| minus(x, s(y)) | → | pred(minus(x, y)) | | quot(0, s(y)) | → | 0 |
| quot(s(x), s(y)) | → | s(quot(minus(x, y), s(y))) |
Original Signature
Termination of terms over the following signature is verified: minus, 0, s, pred, quot
Strategy
Polynomial Interpretation
- 0: 1
- minus(x,y): 2x
- pred(x): x
- quot(x,y): 0
- quot#(x,y): 3x
- s(x): 3x + 1
Improved Usable rules
| pred(s(x)) | → | x | | minus(x, 0) | → | x |
| minus(x, s(y)) | → | pred(minus(x, y)) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| quot#(s(x), s(y)) | → | quot#(minus(x, y), s(y)) |