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 (249ms).
| – Problem 2 remains open; application of the following processors failed [SubtermCriterion (1ms), DependencyGraph (130ms), PolynomialLinearRange4iUR (8146ms), DependencyGraph (100ms), PolynomialLinearRange8NegiUR (30005ms), DependencyGraph (timeout), ReductionPairSAT (3030ms), DependencyGraph (95ms), SizeChangePrinciple (timeout)].
The following open problems remain:
Open Dependency Pair Problem 2
Dependency Pairs
| add#(s(X), Y) | → | activate#(X) | | dbl#(s(X)) | → | activate#(X) |
| activate#(n__add(X1, X2)) | → | activate#(X2) | | activate#(n__dbl(X)) | → | dbl#(activate(X)) |
| terms#(N) | → | sqr#(N) | | activate#(n__first(X1, X2)) | → | activate#(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) | | activate#(n__add(X1, X2)) | → | activate#(X1) |
| first#(s(X), cons(Y, Z)) | → | activate#(Z) | | activate#(n__terms(X)) | → | activate#(X) |
| sqr#(s(X)) | → | activate#(X) | | activate#(n__sqr(X)) | → | activate#(X) |
| activate#(n__sqr(X)) | → | sqr#(activate(X)) | | activate#(n__terms(X)) | → | terms#(activate(X)) |
| activate#(n__add(X1, X2)) | → | add#(activate(X1), activate(X2)) | | activate#(n__dbl(X)) | → | activate#(X) |
Rewrite Rules
| terms(N) | → | cons(recip(sqr(N)), n__terms(n__s(N))) | | sqr(0) | → | 0 |
| sqr(s(X)) | → | s(n__add(n__sqr(activate(X)), n__dbl(activate(X)))) | | dbl(0) | → | 0 |
| dbl(s(X)) | → | s(n__s(n__dbl(activate(X)))) | | add(0, X) | → | X |
| add(s(X), Y) | → | s(n__add(activate(X), Y)) | | first(0, X) | → | nil |
| first(s(X), cons(Y, Z)) | → | cons(Y, n__first(activate(X), activate(Z))) | | terms(X) | → | n__terms(X) |
| s(X) | → | n__s(X) | | add(X1, X2) | → | n__add(X1, X2) |
| sqr(X) | → | n__sqr(X) | | dbl(X) | → | n__dbl(X) |
| first(X1, X2) | → | n__first(X1, X2) | | activate(n__terms(X)) | → | terms(activate(X)) |
| activate(n__s(X)) | → | s(X) | | activate(n__add(X1, X2)) | → | add(activate(X1), activate(X2)) |
| activate(n__sqr(X)) | → | sqr(activate(X)) | | activate(n__dbl(X)) | → | dbl(activate(X)) |
| activate(n__first(X1, X2)) | → | first(activate(X1), activate(X2)) | | activate(X) | → | X |
Original Signature
Termination of terms over the following signature is verified: terms, sqr, n__sqr, dbl, recip, n__terms, add, n__s, activate, 0, s, n__first, n__add, first, n__dbl, nil, cons
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| dbl#(s(X)) | → | s#(n__s(n__dbl(activate(X)))) | | activate#(n__add(X1, X2)) | → | activate#(X2) |
| dbl#(s(X)) | → | activate#(X) | | add#(s(X), Y) | → | activate#(X) |
| activate#(n__dbl(X)) | → | dbl#(activate(X)) | | terms#(N) | → | sqr#(N) |
| sqr#(s(X)) | → | s#(n__add(n__sqr(activate(X)), n__dbl(activate(X)))) | | activate#(n__first(X1, X2)) | → | activate#(X2) |
| first#(s(X), cons(Y, Z)) | → | activate#(X) | | activate#(n__s(X)) | → | s#(X) |
| activate#(n__first(X1, X2)) | → | first#(activate(X1), activate(X2)) | | activate#(n__first(X1, X2)) | → | activate#(X1) |
| add#(s(X), Y) | → | s#(n__add(activate(X), Y)) | | activate#(n__add(X1, X2)) | → | activate#(X1) |
| activate#(n__terms(X)) | → | activate#(X) | | first#(s(X), cons(Y, Z)) | → | activate#(Z) |
| sqr#(s(X)) | → | activate#(X) | | activate#(n__sqr(X)) | → | sqr#(activate(X)) |
| activate#(n__sqr(X)) | → | activate#(X) | | activate#(n__terms(X)) | → | terms#(activate(X)) |
| activate#(n__add(X1, X2)) | → | add#(activate(X1), activate(X2)) | | activate#(n__dbl(X)) | → | activate#(X) |
Rewrite Rules
| terms(N) | → | cons(recip(sqr(N)), n__terms(n__s(N))) | | sqr(0) | → | 0 |
| sqr(s(X)) | → | s(n__add(n__sqr(activate(X)), n__dbl(activate(X)))) | | dbl(0) | → | 0 |
| dbl(s(X)) | → | s(n__s(n__dbl(activate(X)))) | | add(0, X) | → | X |
| add(s(X), Y) | → | s(n__add(activate(X), Y)) | | first(0, X) | → | nil |
| first(s(X), cons(Y, Z)) | → | cons(Y, n__first(activate(X), activate(Z))) | | terms(X) | → | n__terms(X) |
| s(X) | → | n__s(X) | | add(X1, X2) | → | n__add(X1, X2) |
| sqr(X) | → | n__sqr(X) | | dbl(X) | → | n__dbl(X) |
| first(X1, X2) | → | n__first(X1, X2) | | activate(n__terms(X)) | → | terms(activate(X)) |
| activate(n__s(X)) | → | s(X) | | activate(n__add(X1, X2)) | → | add(activate(X1), activate(X2)) |
| activate(n__sqr(X)) | → | sqr(activate(X)) | | activate(n__dbl(X)) | → | dbl(activate(X)) |
| activate(n__first(X1, X2)) | → | first(activate(X1), activate(X2)) | | activate(X) | → | X |
Original Signature
Termination of terms over the following signature is verified: terms, sqr, n__sqr, dbl, recip, n__terms, add, n__s, activate, 0, s, n__first, n__add, n__dbl, first, cons, nil
Strategy
The following SCCs where found
| activate#(n__add(X1, X2)) → activate#(X2) | dbl#(s(X)) → activate#(X) |
| add#(s(X), Y) → activate#(X) | activate#(n__dbl(X)) → dbl#(activate(X)) |
| terms#(N) → sqr#(N) | activate#(n__first(X1, X2)) → activate#(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) | activate#(n__add(X1, X2)) → activate#(X1) |
| activate#(n__terms(X)) → activate#(X) | first#(s(X), cons(Y, Z)) → activate#(Z) |
| sqr#(s(X)) → activate#(X) | activate#(n__sqr(X)) → sqr#(activate(X)) |
| activate#(n__sqr(X)) → activate#(X) | activate#(n__terms(X)) → terms#(activate(X)) |
| activate#(n__add(X1, X2)) → add#(activate(X1), activate(X2)) | activate#(n__dbl(X)) → activate#(X) |