YES
The TRS could be proven terminating. The proof took 60000 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (33ms).
| – Problem 2 was processed with processor SubtermCriterion (4ms).
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| activate#(n__cons(X1, X2)) | → | cons#(activate(X1), X2) | | activate#(n__cons(X1, X2)) | → | activate#(X1) |
| from#(X) | → | cons#(X, n__from(n__s(X))) | | activate#(n__from(X)) | → | from#(activate(X)) |
| activate#(n__s(X)) | → | activate#(X) | | activate#(n__from(X)) | → | activate#(X) |
| 2nd#(cons(X, n__cons(Y, Z))) | → | activate#(Y) | | activate#(n__s(X)) | → | s#(activate(X)) |
Rewrite Rules
| 2nd(cons(X, n__cons(Y, Z))) | → | activate(Y) | | from(X) | → | cons(X, n__from(n__s(X))) |
| cons(X1, X2) | → | n__cons(X1, X2) | | from(X) | → | n__from(X) |
| s(X) | → | n__s(X) | | activate(n__cons(X1, X2)) | → | cons(activate(X1), X2) |
| activate(n__from(X)) | → | from(activate(X)) | | activate(n__s(X)) | → | s(activate(X)) |
| activate(X) | → | X |
Original Signature
Termination of terms over the following signature is verified: activate, 2nd, n__s, n__cons, n__from, s, from, cons
Strategy
The following SCCs where found
| activate#(n__cons(X1, X2)) → activate#(X1) | activate#(n__s(X)) → activate#(X) |
| activate#(n__from(X)) → activate#(X) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| activate#(n__cons(X1, X2)) | → | activate#(X1) | | activate#(n__s(X)) | → | activate#(X) |
| activate#(n__from(X)) | → | activate#(X) |
Rewrite Rules
| 2nd(cons(X, n__cons(Y, Z))) | → | activate(Y) | | from(X) | → | cons(X, n__from(n__s(X))) |
| cons(X1, X2) | → | n__cons(X1, X2) | | from(X) | → | n__from(X) |
| s(X) | → | n__s(X) | | activate(n__cons(X1, X2)) | → | cons(activate(X1), X2) |
| activate(n__from(X)) | → | from(activate(X)) | | activate(n__s(X)) | → | s(activate(X)) |
| activate(X) | → | X |
Original Signature
Termination of terms over the following signature is verified: activate, 2nd, n__s, n__cons, n__from, s, from, cons
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| activate#(n__cons(X1, X2)) | → | activate#(X1) | | activate#(n__s(X)) | → | activate#(X) |
| activate#(n__from(X)) | → | activate#(X) |