MAYBE
The TRS could not be proven terminating. The proof attempt took 475 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (0ms).
| – Problem 2 was processed with processor SubtermCriterion (0ms).
| – Problem 3 was processed with processor SubtermCriterion (0ms).
| – Problem 4 remains open; application of the following processors failed [SubtermCriterion (0ms), DependencyGraph (1ms), PolynomialLinearRange4iUR (101ms), DependencyGraph (0ms), PolynomialLinearRange8NegiUR (165ms), DependencyGraph (1ms), ReductionPairSAT (75ms), DependencyGraph (1ms), SizeChangePrinciple (5ms)].
The following open problems remain:
Open Dependency Pair Problem 4
Dependency Pairs
Rewrite Rules
| f(X) | → | cons(X, f(g(X))) | | g(0) | → | s(0) |
| g(s(X)) | → | s(s(g(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: f, g, 0, s, sel, cons
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| g#(s(X)) | → | g#(X) | | sel#(s(X), cons(Y, Z)) | → | sel#(X, Z) |
| f#(X) | → | g#(X) | | f#(X) | → | f#(g(X)) |
Rewrite Rules
| f(X) | → | cons(X, f(g(X))) | | g(0) | → | s(0) |
| g(s(X)) | → | s(s(g(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: f, g, 0, s, sel, cons
Strategy
The following SCCs where found
| sel#(s(X), cons(Y, Z)) → sel#(X, Z) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| sel#(s(X), cons(Y, Z)) | → | sel#(X, Z) |
Rewrite Rules
| f(X) | → | cons(X, f(g(X))) | | g(0) | → | s(0) |
| g(s(X)) | → | s(s(g(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: f, g, 0, s, 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) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
Rewrite Rules
| f(X) | → | cons(X, f(g(X))) | | g(0) | → | s(0) |
| g(s(X)) | → | s(s(g(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: f, g, 0, s, sel, cons
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed: