MAYBE
The TRS could not be proven terminating. The proof attempt took 780 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (0ms).
| – Problem 2 remains open; application of the following processors failed [SubtermCriterion (1ms), DependencyGraph (3ms), PolynomialLinearRange4iUR (139ms), DependencyGraph (4ms), PolynomialLinearRange8NegiUR (333ms), DependencyGraph (2ms), ReductionPairSAT (128ms), DependencyGraph (3ms), SizeChangePrinciple (17ms)].
| – Problem 3 was processed with processor SubtermCriterion (0ms).
The following open problems remain:
Open Dependency Pair Problem 2
Dependency Pairs
| if#(true, x, y) | → | int#(s(x), y) | | int#(x, y) | → | if#(le(x, y), x, y) |
Rewrite Rules
| le(0, y) | → | true | | le(s(x), 0) | → | false |
| le(s(x), s(y)) | → | le(x, y) | | int(x, y) | → | if(le(x, y), x, y) |
| if(true, x, y) | → | cons(x, int(s(x), y)) | | if(false, x, y) | → | nil |
Original Signature
Termination of terms over the following signature is verified: 0, s, le, int, if, false, true, nil, cons
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| if#(true, x, y) | → | int#(s(x), y) | | le#(s(x), s(y)) | → | le#(x, y) |
| int#(x, y) | → | if#(le(x, y), x, y) | | int#(x, y) | → | le#(x, y) |
Rewrite Rules
| le(0, y) | → | true | | le(s(x), 0) | → | false |
| le(s(x), s(y)) | → | le(x, y) | | int(x, y) | → | if(le(x, y), x, y) |
| if(true, x, y) | → | cons(x, int(s(x), y)) | | if(false, x, y) | → | nil |
Original Signature
Termination of terms over the following signature is verified: 0, le, s, if, int, true, false, cons, nil
Strategy
The following SCCs where found
| if#(true, x, y) → int#(s(x), y) | int#(x, y) → if#(le(x, y), x, y) |
| le#(s(x), s(y)) → le#(x, y) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| le#(s(x), s(y)) | → | le#(x, y) |
Rewrite Rules
| le(0, y) | → | true | | le(s(x), 0) | → | false |
| le(s(x), s(y)) | → | le(x, y) | | int(x, y) | → | if(le(x, y), x, y) |
| if(true, x, y) | → | cons(x, int(s(x), y)) | | if(false, x, y) | → | nil |
Original Signature
Termination of terms over the following signature is verified: 0, le, s, if, int, true, false, cons, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| le#(s(x), s(y)) | → | le#(x, y) |