TIMEOUT
The TRS could not be proven terminating. The proof attempt took 60000 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (135ms).
| – Problem 2 remains open; application of the following processors failed [SubtermCriterion (1ms), DependencyGraph (54ms), PolynomialLinearRange4iUR (1384ms), DependencyGraph (52ms), PolynomialLinearRange8NegiUR (12113ms), DependencyGraph (42ms), ReductionPairSAT (1282ms), DependencyGraph (42ms), SizeChangePrinciple (timeout)].
The following open problems remain:
Open Dependency Pair Problem 2
Dependency Pairs
| add#(s(X), Y) | → | activate#(X) | | add#(s(X), Y) | → | activate#(Y) |
| activate#(n__from(X)) | → | from#(X) | | activate#(n__first(X1, X2)) | → | activate#(X2) |
| activate#(n__add(X1, X2)) | → | add#(activate(X1), X2) | | first#(s(X), cons(Y, Z)) | → | activate#(X) |
| activate#(n__first(X1, X2)) | → | first#(activate(X1), activate(X2)) | | activate#(n__first(X1, X2)) | → | activate#(X1) |
| from#(X) | → | activate#(X) | | activate#(n__add(X1, X2)) | → | activate#(X1) |
| first#(s(X), cons(Y, Z)) | → | activate#(Z) | | first#(s(X), cons(Y, Z)) | → | activate#(Y) |
| add#(0, X) | → | activate#(X) |
Rewrite Rules
| and(true, X) | → | activate(X) | | and(false, Y) | → | false |
| if(true, X, Y) | → | activate(X) | | if(false, X, Y) | → | activate(Y) |
| add(0, X) | → | activate(X) | | add(s(X), Y) | → | s(n__add(activate(X), activate(Y))) |
| first(0, X) | → | nil | | first(s(X), cons(Y, Z)) | → | cons(activate(Y), n__first(activate(X), activate(Z))) |
| from(X) | → | cons(activate(X), n__from(n__s(activate(X)))) | | add(X1, X2) | → | n__add(X1, X2) |
| first(X1, X2) | → | n__first(X1, X2) | | from(X) | → | n__from(X) |
| s(X) | → | n__s(X) | | activate(n__add(X1, X2)) | → | add(activate(X1), X2) |
| activate(n__first(X1, X2)) | → | first(activate(X1), activate(X2)) | | activate(n__from(X)) | → | from(X) |
| activate(n__s(X)) | → | s(X) | | activate(X) | → | X |
Original Signature
Termination of terms over the following signature is verified: n__from, true, from, add, and, n__s, activate, 0, s, n__first, if, n__add, false, first, nil, cons
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| add#(s(X), Y) | → | activate#(X) | | add#(s(X), Y) | → | activate#(Y) |
| activate#(n__from(X)) | → | from#(X) | | activate#(n__first(X1, X2)) | → | activate#(X2) |
| activate#(n__s(X)) | → | s#(X) | | add#(s(X), Y) | → | s#(n__add(activate(X), activate(Y))) |
| first#(s(X), cons(Y, Z)) | → | activate#(X) | | activate#(n__add(X1, X2)) | → | add#(activate(X1), X2) |
| activate#(n__first(X1, X2)) | → | first#(activate(X1), activate(X2)) | | activate#(n__first(X1, X2)) | → | activate#(X1) |
| from#(X) | → | activate#(X) | | activate#(n__add(X1, X2)) | → | activate#(X1) |
| first#(s(X), cons(Y, Z)) | → | activate#(Z) | | if#(false, X, Y) | → | activate#(Y) |
| if#(true, X, Y) | → | activate#(X) | | and#(true, X) | → | activate#(X) |
| first#(s(X), cons(Y, Z)) | → | activate#(Y) | | add#(0, X) | → | activate#(X) |
Rewrite Rules
| and(true, X) | → | activate(X) | | and(false, Y) | → | false |
| if(true, X, Y) | → | activate(X) | | if(false, X, Y) | → | activate(Y) |
| add(0, X) | → | activate(X) | | add(s(X), Y) | → | s(n__add(activate(X), activate(Y))) |
| first(0, X) | → | nil | | first(s(X), cons(Y, Z)) | → | cons(activate(Y), n__first(activate(X), activate(Z))) |
| from(X) | → | cons(activate(X), n__from(n__s(activate(X)))) | | add(X1, X2) | → | n__add(X1, X2) |
| first(X1, X2) | → | n__first(X1, X2) | | from(X) | → | n__from(X) |
| s(X) | → | n__s(X) | | activate(n__add(X1, X2)) | → | add(activate(X1), X2) |
| activate(n__first(X1, X2)) | → | first(activate(X1), activate(X2)) | | activate(n__from(X)) | → | from(X) |
| activate(n__s(X)) | → | s(X) | | activate(X) | → | X |
Original Signature
Termination of terms over the following signature is verified: n__from, true, from, add, and, n__s, activate, 0, s, n__first, if, n__add, false, first, cons, nil
Strategy
The following SCCs where found
| add#(s(X), Y) → activate#(X) | add#(s(X), Y) → activate#(Y) |
| activate#(n__from(X)) → from#(X) | activate#(n__first(X1, X2)) → activate#(X2) |
| first#(s(X), cons(Y, Z)) → activate#(X) | activate#(n__add(X1, X2)) → add#(activate(X1), X2) |
| activate#(n__first(X1, X2)) → first#(activate(X1), activate(X2)) | activate#(n__first(X1, X2)) → activate#(X1) |
| from#(X) → activate#(X) | activate#(n__add(X1, X2)) → activate#(X1) |
| first#(s(X), cons(Y, Z)) → activate#(Z) | add#(0, X) → activate#(X) |
| first#(s(X), cons(Y, Z)) → activate#(Y) |