MAYBE
The TRS could not be proven terminating. The proof attempt took 279 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 (2ms), DependencyGraph (1ms), PolynomialLinearRange4iUR (59ms), DependencyGraph (1ms), PolynomialLinearRange8NegiUR (35ms), DependencyGraph (1ms), ReductionPairSAT (21ms), DependencyGraph (1ms), SizeChangePrinciple (4ms)].
| – Problem 3 was processed with processor SubtermCriterion (0ms).
The following open problems remain:
Open Dependency Pair Problem 2
Dependency Pairs
Rewrite Rules
| from(X) | → | cons(X, from(s(X))) | | sel(0, cons(X, Y)) | → | X |
| sel(s(X), cons(Y, Z)) | → | sel(X, Z) |
Original Signature
Termination of terms over the following signature is verified: 0, s, from, sel, cons
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| sel#(s(X), cons(Y, Z)) | → | sel#(X, Z) | | from#(X) | → | from#(s(X)) |
Rewrite Rules
| from(X) | → | cons(X, from(s(X))) | | sel(0, cons(X, Y)) | → | X |
| sel(s(X), cons(Y, Z)) | → | sel(X, Z) |
Original Signature
Termination of terms over the following signature is verified: 0, s, from, sel, cons
Strategy
The following SCCs where found
| sel#(s(X), cons(Y, Z)) → sel#(X, Z) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| sel#(s(X), cons(Y, Z)) | → | sel#(X, Z) |
Rewrite Rules
| from(X) | → | cons(X, from(s(X))) | | sel(0, cons(X, Y)) | → | X |
| sel(s(X), cons(Y, Z)) | → | sel(X, Z) |
Original Signature
Termination of terms over the following signature is verified: 0, s, from, sel, cons
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| sel#(s(X), cons(Y, Z)) | → | sel#(X, Z) |