YES
The TRS could be proven terminating. The proof took 341 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (7ms).
| – Problem 2 was processed with processor PolynomialLinearRange4iUR (202ms).
| – Problem 3 was processed with processor SubtermCriterion (0ms).
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| remove#(x, .(y, z)) | → | remove#(x, z) | | purge#(.(x, y)) | → | remove#(x, y) |
| purge#(.(x, y)) | → | purge#(remove(x, y)) |
Rewrite Rules
| purge(nil) | → | nil | | purge(.(x, y)) | → | .(x, purge(remove(x, y))) |
| remove(x, nil) | → | nil | | remove(x, .(y, z)) | → | if(=(x, y), remove(x, z), .(y, remove(x, z))) |
Original Signature
Termination of terms over the following signature is verified: remove, if, ., purge, =, nil
Strategy
The following SCCs where found
| remove#(x, .(y, z)) → remove#(x, z) |
| purge#(.(x, y)) → purge#(remove(x, y)) |
Problem 2: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| purge#(.(x, y)) | → | purge#(remove(x, y)) |
Rewrite Rules
| purge(nil) | → | nil | | purge(.(x, y)) | → | .(x, purge(remove(x, y))) |
| remove(x, nil) | → | nil | | remove(x, .(y, z)) | → | if(=(x, y), remove(x, z), .(y, remove(x, z))) |
Original Signature
Termination of terms over the following signature is verified: remove, if, ., purge, =, nil
Strategy
Polynomial Interpretation
- .(x,y): 1
- =(x,y): 3
- if(x,y,z): 0
- nil: 0
- purge(x): 0
- purge#(x): 2x
- remove(x,y): 0
Improved Usable rules
| remove(x, nil) | → | nil | | remove(x, .(y, z)) | → | if(=(x, y), remove(x, z), .(y, remove(x, z))) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| purge#(.(x, y)) | → | purge#(remove(x, y)) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| remove#(x, .(y, z)) | → | remove#(x, z) |
Rewrite Rules
| purge(nil) | → | nil | | purge(.(x, y)) | → | .(x, purge(remove(x, y))) |
| remove(x, nil) | → | nil | | remove(x, .(y, z)) | → | if(=(x, y), remove(x, z), .(y, remove(x, z))) |
Original Signature
Termination of terms over the following signature is verified: remove, if, ., purge, =, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| remove#(x, .(y, z)) | → | remove#(x, z) |