YES
The TRS could be proven terminating. The proof took 496 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (14ms).
| – Problem 2 was processed with processor PolynomialLinearRange4iUR (331ms).
| – Problem 3 was processed with processor SubtermCriterion (1ms).
| – Problem 4 was processed with processor SubtermCriterion (1ms).
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| lowers#(x, .(y, z)) | → | lowers#(x, z) | | greaters#(x, .(y, z)) | → | greaters#(x, z) |
| qsort#(.(x, y)) | → | qsort#(greaters(x, y)) | | qsort#(.(x, y)) | → | qsort#(lowers(x, y)) |
| qsort#(.(x, y)) | → | greaters#(x, y) | | qsort#(.(x, y)) | → | lowers#(x, y) |
Rewrite Rules
| qsort(nil) | → | nil | | qsort(.(x, y)) | → | ++(qsort(lowers(x, y)), .(x, qsort(greaters(x, y)))) |
| lowers(x, nil) | → | nil | | lowers(x, .(y, z)) | → | if(<=(y, x), .(y, lowers(x, z)), lowers(x, z)) |
| greaters(x, nil) | → | nil | | greaters(x, .(y, z)) | → | if(<=(y, x), greaters(x, z), .(y, greaters(x, z))) |
Original Signature
Termination of terms over the following signature is verified: if, <=, qsort, ++, ., nil, lowers, greaters
Strategy
The following SCCs where found
| lowers#(x, .(y, z)) → lowers#(x, z) |
| qsort#(.(x, y)) → qsort#(greaters(x, y)) | qsort#(.(x, y)) → qsort#(lowers(x, y)) |
| greaters#(x, .(y, z)) → greaters#(x, z) |
Problem 2: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| qsort#(.(x, y)) | → | qsort#(greaters(x, y)) | | qsort#(.(x, y)) | → | qsort#(lowers(x, y)) |
Rewrite Rules
| qsort(nil) | → | nil | | qsort(.(x, y)) | → | ++(qsort(lowers(x, y)), .(x, qsort(greaters(x, y)))) |
| lowers(x, nil) | → | nil | | lowers(x, .(y, z)) | → | if(<=(y, x), .(y, lowers(x, z)), lowers(x, z)) |
| greaters(x, nil) | → | nil | | greaters(x, .(y, z)) | → | if(<=(y, x), greaters(x, z), .(y, greaters(x, z))) |
Original Signature
Termination of terms over the following signature is verified: if, <=, qsort, ++, ., nil, lowers, greaters
Strategy
Polynomial Interpretation
- ++(x,y): 0
- .(x,y): 2y + 2x + 1
- <=(x,y): 3
- greaters(x,y): x
- if(x,y,z): 0
- lowers(x,y): 2x
- nil: 0
- qsort(x): 0
- qsort#(x): x
Improved Usable rules
| greaters(x, .(y, z)) | → | if(<=(y, x), greaters(x, z), .(y, greaters(x, z))) | | greaters(x, nil) | → | nil |
| lowers(x, nil) | → | nil | | lowers(x, .(y, z)) | → | if(<=(y, x), .(y, lowers(x, z)), lowers(x, z)) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| qsort#(.(x, y)) | → | qsort#(greaters(x, y)) | | qsort#(.(x, y)) | → | qsort#(lowers(x, y)) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| greaters#(x, .(y, z)) | → | greaters#(x, z) |
Rewrite Rules
| qsort(nil) | → | nil | | qsort(.(x, y)) | → | ++(qsort(lowers(x, y)), .(x, qsort(greaters(x, y)))) |
| lowers(x, nil) | → | nil | | lowers(x, .(y, z)) | → | if(<=(y, x), .(y, lowers(x, z)), lowers(x, z)) |
| greaters(x, nil) | → | nil | | greaters(x, .(y, z)) | → | if(<=(y, x), greaters(x, z), .(y, greaters(x, z))) |
Original Signature
Termination of terms over the following signature is verified: if, <=, qsort, ++, ., nil, lowers, greaters
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| greaters#(x, .(y, z)) | → | greaters#(x, z) |
Problem 4: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| lowers#(x, .(y, z)) | → | lowers#(x, z) |
Rewrite Rules
| qsort(nil) | → | nil | | qsort(.(x, y)) | → | ++(qsort(lowers(x, y)), .(x, qsort(greaters(x, y)))) |
| lowers(x, nil) | → | nil | | lowers(x, .(y, z)) | → | if(<=(y, x), .(y, lowers(x, z)), lowers(x, z)) |
| greaters(x, nil) | → | nil | | greaters(x, .(y, z)) | → | if(<=(y, x), greaters(x, z), .(y, greaters(x, z))) |
Original Signature
Termination of terms over the following signature is verified: if, <=, qsort, ++, ., nil, lowers, greaters
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| lowers#(x, .(y, z)) | → | lowers#(x, z) |