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 (178ms).
| – Problem 2 remains open; application of the following processors failed [SubtermCriterion (1ms), DependencyGraph (60ms), PolynomialLinearRange4iUR (2723ms), DependencyGraph (61ms), PolynomialLinearRange8NegiUR (17199ms), DependencyGraph (47ms), ReductionPairSAT (3802ms), DependencyGraph (71ms), SizeChangePrinciple (timeout)].
The following open problems remain:
Open Dependency Pair Problem 2
Dependency Pairs
| zWadr#(cons(X, XS), cons(Y, YS)) | → | activate#(YS) | | app#(cons(X, XS), YS) | → | activate#(XS) |
| activate#(n__s(X)) | → | activate#(X) | | zWadr#(cons(X, XS), cons(Y, YS)) | → | app#(Y, cons(X, n__nil)) |
| activate#(n__prefix(X)) | → | activate#(X) | | activate#(n__from(X)) | → | activate#(X) |
| activate#(n__app(X1, X2)) | → | activate#(X1) | | activate#(n__zWadr(X1, X2)) | → | activate#(X2) |
| activate#(n__zWadr(X1, X2)) | → | zWadr#(activate(X1), activate(X2)) | | activate#(n__zWadr(X1, X2)) | → | activate#(X1) |
| activate#(n__app(X1, X2)) | → | app#(activate(X1), activate(X2)) | | zWadr#(cons(X, XS), cons(Y, YS)) | → | activate#(XS) |
| activate#(n__app(X1, X2)) | → | activate#(X2) |
Rewrite Rules
| app(nil, YS) | → | YS | | app(cons(X, XS), YS) | → | cons(X, n__app(activate(XS), YS)) |
| from(X) | → | cons(X, n__from(n__s(X))) | | zWadr(nil, YS) | → | nil |
| zWadr(XS, nil) | → | nil | | zWadr(cons(X, XS), cons(Y, YS)) | → | cons(app(Y, cons(X, n__nil)), n__zWadr(activate(XS), activate(YS))) |
| prefix(L) | → | cons(nil, n__zWadr(L, n__prefix(L))) | | app(X1, X2) | → | n__app(X1, X2) |
| from(X) | → | n__from(X) | | s(X) | → | n__s(X) |
| nil | → | n__nil | | zWadr(X1, X2) | → | n__zWadr(X1, X2) |
| prefix(X) | → | n__prefix(X) | | activate(n__app(X1, X2)) | → | app(activate(X1), activate(X2)) |
| activate(n__from(X)) | → | from(activate(X)) | | activate(n__s(X)) | → | s(activate(X)) |
| activate(n__nil) | → | nil | | activate(n__zWadr(X1, X2)) | → | zWadr(activate(X1), activate(X2)) |
| activate(n__prefix(X)) | → | prefix(activate(X)) | | activate(X) | → | X |
Original Signature
Termination of terms over the following signature is verified: app, n__from, n__app, from, n__prefix, activate, n__s, zWadr, s, n__zWadr, prefix, n__nil, nil, cons
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| prefix#(L) | → | nil# | | zWadr#(cons(X, XS), cons(Y, YS)) | → | activate#(YS) |
| app#(cons(X, XS), YS) | → | activate#(XS) | | activate#(n__s(X)) | → | activate#(X) |
| zWadr#(cons(X, XS), cons(Y, YS)) | → | app#(Y, cons(X, n__nil)) | | activate#(n__prefix(X)) | → | activate#(X) |
| activate#(n__from(X)) | → | activate#(X) | | activate#(n__app(X1, X2)) | → | activate#(X1) |
| activate#(n__zWadr(X1, X2)) | → | activate#(X2) | | zWadr#(nil, YS) | → | nil# |
| activate#(n__zWadr(X1, X2)) | → | zWadr#(activate(X1), activate(X2)) | | activate#(n__nil) | → | nil# |
| activate#(n__prefix(X)) | → | prefix#(activate(X)) | | activate#(n__app(X1, X2)) | → | app#(activate(X1), activate(X2)) |
| activate#(n__zWadr(X1, X2)) | → | activate#(X1) | | zWadr#(XS, nil) | → | nil# |
| activate#(n__from(X)) | → | from#(activate(X)) | | zWadr#(cons(X, XS), cons(Y, YS)) | → | activate#(XS) |
| activate#(n__s(X)) | → | s#(activate(X)) | | activate#(n__app(X1, X2)) | → | activate#(X2) |
Rewrite Rules
| app(nil, YS) | → | YS | | app(cons(X, XS), YS) | → | cons(X, n__app(activate(XS), YS)) |
| from(X) | → | cons(X, n__from(n__s(X))) | | zWadr(nil, YS) | → | nil |
| zWadr(XS, nil) | → | nil | | zWadr(cons(X, XS), cons(Y, YS)) | → | cons(app(Y, cons(X, n__nil)), n__zWadr(activate(XS), activate(YS))) |
| prefix(L) | → | cons(nil, n__zWadr(L, n__prefix(L))) | | app(X1, X2) | → | n__app(X1, X2) |
| from(X) | → | n__from(X) | | s(X) | → | n__s(X) |
| nil | → | n__nil | | zWadr(X1, X2) | → | n__zWadr(X1, X2) |
| prefix(X) | → | n__prefix(X) | | activate(n__app(X1, X2)) | → | app(activate(X1), activate(X2)) |
| activate(n__from(X)) | → | from(activate(X)) | | activate(n__s(X)) | → | s(activate(X)) |
| activate(n__nil) | → | nil | | activate(n__zWadr(X1, X2)) | → | zWadr(activate(X1), activate(X2)) |
| activate(n__prefix(X)) | → | prefix(activate(X)) | | activate(X) | → | X |
Original Signature
Termination of terms over the following signature is verified: app, n__from, n__app, from, n__prefix, activate, n__s, zWadr, s, n__zWadr, prefix, n__nil, cons, nil
Strategy
The following SCCs where found
| zWadr#(cons(X, XS), cons(Y, YS)) → activate#(YS) | app#(cons(X, XS), YS) → activate#(XS) |
| activate#(n__s(X)) → activate#(X) | zWadr#(cons(X, XS), cons(Y, YS)) → app#(Y, cons(X, n__nil)) |
| activate#(n__from(X)) → activate#(X) | activate#(n__prefix(X)) → activate#(X) |
| activate#(n__app(X1, X2)) → activate#(X1) | activate#(n__zWadr(X1, X2)) → activate#(X2) |
| activate#(n__zWadr(X1, X2)) → zWadr#(activate(X1), activate(X2)) | activate#(n__app(X1, X2)) → app#(activate(X1), activate(X2)) |
| activate#(n__zWadr(X1, X2)) → activate#(X1) | zWadr#(cons(X, XS), cons(Y, YS)) → activate#(XS) |
| activate#(n__app(X1, X2)) → activate#(X2) |