MAYBE
The TRS could not be proven terminating. The proof attempt took 265 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 (1ms), PolynomialLinearRange4iUR (57ms), DependencyGraph (1ms), PolynomialLinearRange8NegiUR (56ms), DependencyGraph (0ms), ReductionPairSAT (18ms), DependencyGraph (1ms), SizeChangePrinciple (4ms)].
| – Problem 3 was processed with processor SubtermCriterion (0ms).
| | – Problem 4 was processed with processor DependencyGraph (0ms).
The following open problems remain:
Open Dependency Pair Problem 2
Dependency Pairs
Rewrite Rules
| from(X) | → | cons(X, from(s(X))) | | length(nil) | → | 0 |
| length(cons(X, Y)) | → | s(length1(Y)) | | length1(X) | → | length(X) |
Original Signature
Termination of terms over the following signature is verified: 0, s, length, length1, from, nil, cons
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| from#(X) | → | from#(s(X)) | | length1#(X) | → | length#(X) |
| length#(cons(X, Y)) | → | length1#(Y) |
Rewrite Rules
| from(X) | → | cons(X, from(s(X))) | | length(nil) | → | 0 |
| length(cons(X, Y)) | → | s(length1(Y)) | | length1(X) | → | length(X) |
Original Signature
Termination of terms over the following signature is verified: 0, s, length, from, length1, cons, nil
Strategy
The following SCCs where found
| length1#(X) → length#(X) | length#(cons(X, Y)) → length1#(Y) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| length1#(X) | → | length#(X) | | length#(cons(X, Y)) | → | length1#(Y) |
Rewrite Rules
| from(X) | → | cons(X, from(s(X))) | | length(nil) | → | 0 |
| length(cons(X, Y)) | → | s(length1(Y)) | | length1(X) | → | length(X) |
Original Signature
Termination of terms over the following signature is verified: 0, s, length, from, length1, cons, nil
Strategy
Projection
The following projection was used:
- π (length#): 1
- π (length1#): 1
Thus, the following dependency pairs are removed:
| length#(cons(X, Y)) | → | length1#(Y) |
Problem 4: DependencyGraph
Dependency Pair Problem
Dependency Pairs
Rewrite Rules
| from(X) | → | cons(X, from(s(X))) | | length(nil) | → | 0 |
| length(cons(X, Y)) | → | s(length1(Y)) | | length1(X) | → | length(X) |
Original Signature
Termination of terms over the following signature is verified: 0, s, length, length1, from, nil, cons
Strategy
There are no SCCs!