YES
The TRS could be proven terminating. The proof took 43091 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (739ms).
| – Problem 2 was processed with processor SubtermCriterion (0ms).
| | – Problem 9 was processed with processor SubtermCriterion (0ms).
| – Problem 3 was processed with processor SubtermCriterion (1ms).
| | – Problem 10 was processed with processor SubtermCriterion (1ms).
| – Problem 4 was processed with processor SubtermCriterion (0ms).
| – Problem 5 was processed with processor PolynomialLinearRange4 (295ms).
| | – Problem 12 was processed with processor PolynomialLinearRange4 (349ms).
| | | – Problem 13 was processed with processor PolynomialLinearRange4 (237ms).
| | | | – Problem 14 was processed with processor PolynomialLinearRange4 (227ms).
| | | | | – Problem 15 was processed with processor PolynomialLinearRange4 (203ms).
| | | | | | – Problem 16 was processed with processor PolynomialLinearRange4 (235ms).
| | | | | | | – Problem 17 was processed with processor PolynomialLinearRange4 (163ms).
| | | | | | | | – Problem 18 was processed with processor PolynomialLinearRange4 (146ms).
| | | | | | | | | – Problem 19 was processed with processor DependencyGraph (2ms).
| | | | | | | | | | – Problem 20 was processed with processor PolynomialLinearRange4 (9ms).
| | | | | | | | | | | – Problem 21 was processed with processor PolynomialLinearRange4 (8ms).
| – Problem 6 was processed with processor SubtermCriterion (1ms).
| – Problem 7 was processed with processor SubtermCriterion (0ms).
| – Problem 8 was processed with processor SubtermCriterion (1ms).
| | – Problem 11 was processed with processor SubtermCriterion (0ms).
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| mark#(cons(X1, X2)) | → | active#(cons(mark(X1), X2)) | | mark#(zWadr(X1, X2)) | → | mark#(X2) |
| active#(app(nil, YS)) | → | mark#(YS) | | mark#(s(X)) | → | s#(mark(X)) |
| app#(mark(X1), X2) | → | app#(X1, X2) | | active#(zWadr(nil, YS)) | → | mark#(nil) |
| active#(prefix(L)) | → | mark#(cons(nil, zWadr(L, prefix(L)))) | | active#(zWadr(cons(X, XS), cons(Y, YS))) | → | cons#(X, nil) |
| active#(prefix(L)) | → | prefix#(L) | | mark#(s(X)) | → | mark#(X) |
| mark#(app(X1, X2)) | → | active#(app(mark(X1), mark(X2))) | | active#(app(cons(X, XS), YS)) | → | cons#(X, app(XS, YS)) |
| active#(zWadr(cons(X, XS), cons(Y, YS))) | → | zWadr#(XS, YS) | | mark#(from(X)) | → | mark#(X) |
| mark#(cons(X1, X2)) | → | mark#(X1) | | mark#(app(X1, X2)) | → | app#(mark(X1), mark(X2)) |
| active#(from(X)) | → | mark#(cons(X, from(s(X)))) | | active#(app(cons(X, XS), YS)) | → | app#(XS, YS) |
| zWadr#(X1, mark(X2)) | → | zWadr#(X1, X2) | | active#(from(X)) | → | s#(X) |
| cons#(X1, active(X2)) | → | cons#(X1, X2) | | active#(zWadr(cons(X, XS), cons(Y, YS))) | → | app#(Y, cons(X, nil)) |
| zWadr#(active(X1), X2) | → | zWadr#(X1, X2) | | mark#(from(X)) | → | from#(mark(X)) |
| prefix#(mark(X)) | → | prefix#(X) | | mark#(prefix(X)) | → | active#(prefix(mark(X))) |
| zWadr#(mark(X1), X2) | → | zWadr#(X1, X2) | | mark#(from(X)) | → | active#(from(mark(X))) |
| active#(zWadr(cons(X, XS), cons(Y, YS))) | → | cons#(app(Y, cons(X, nil)), zWadr(XS, YS)) | | from#(active(X)) | → | from#(X) |
| mark#(zWadr(X1, X2)) | → | zWadr#(mark(X1), mark(X2)) | | mark#(zWadr(X1, X2)) | → | active#(zWadr(mark(X1), mark(X2))) |
| mark#(app(X1, X2)) | → | mark#(X2) | | app#(X1, active(X2)) | → | app#(X1, X2) |
| active#(prefix(L)) | → | zWadr#(L, prefix(L)) | | prefix#(active(X)) | → | prefix#(X) |
| cons#(mark(X1), X2) | → | cons#(X1, X2) | | zWadr#(X1, active(X2)) | → | zWadr#(X1, X2) |
| from#(mark(X)) | → | from#(X) | | mark#(prefix(X)) | → | mark#(X) |
| mark#(nil) | → | active#(nil) | | mark#(app(X1, X2)) | → | mark#(X1) |
| active#(from(X)) | → | cons#(X, from(s(X))) | | cons#(X1, mark(X2)) | → | cons#(X1, X2) |
| app#(X1, mark(X2)) | → | app#(X1, X2) | | mark#(cons(X1, X2)) | → | cons#(mark(X1), X2) |
| mark#(s(X)) | → | active#(s(mark(X))) | | active#(zWadr(XS, nil)) | → | mark#(nil) |
| cons#(active(X1), X2) | → | cons#(X1, X2) | | mark#(prefix(X)) | → | prefix#(mark(X)) |
| s#(mark(X)) | → | s#(X) | | active#(zWadr(cons(X, XS), cons(Y, YS))) | → | mark#(cons(app(Y, cons(X, nil)), zWadr(XS, YS))) |
| active#(prefix(L)) | → | cons#(nil, zWadr(L, prefix(L))) | | mark#(zWadr(X1, X2)) | → | mark#(X1) |
| s#(active(X)) | → | s#(X) | | active#(app(cons(X, XS), YS)) | → | mark#(cons(X, app(XS, YS))) |
| app#(active(X1), X2) | → | app#(X1, X2) | | active#(from(X)) | → | from#(s(X)) |
Rewrite Rules
| active(app(nil, YS)) | → | mark(YS) | | active(app(cons(X, XS), YS)) | → | mark(cons(X, app(XS, YS))) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(zWadr(nil, YS)) | → | mark(nil) |
| active(zWadr(XS, nil)) | → | mark(nil) | | active(zWadr(cons(X, XS), cons(Y, YS))) | → | mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS))) |
| active(prefix(L)) | → | mark(cons(nil, zWadr(L, prefix(L)))) | | mark(app(X1, X2)) | → | active(app(mark(X1), mark(X2))) |
| mark(nil) | → | active(nil) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(from(X)) | → | active(from(mark(X))) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(zWadr(X1, X2)) | → | active(zWadr(mark(X1), mark(X2))) | | mark(prefix(X)) | → | active(prefix(mark(X))) |
| app(mark(X1), X2) | → | app(X1, X2) | | app(X1, mark(X2)) | → | app(X1, X2) |
| app(active(X1), X2) | → | app(X1, X2) | | app(X1, active(X2)) | → | app(X1, X2) |
| cons(mark(X1), X2) | → | cons(X1, X2) | | cons(X1, mark(X2)) | → | cons(X1, X2) |
| cons(active(X1), X2) | → | cons(X1, X2) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| from(mark(X)) | → | from(X) | | from(active(X)) | → | from(X) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| zWadr(mark(X1), X2) | → | zWadr(X1, X2) | | zWadr(X1, mark(X2)) | → | zWadr(X1, X2) |
| zWadr(active(X1), X2) | → | zWadr(X1, X2) | | zWadr(X1, active(X2)) | → | zWadr(X1, X2) |
| prefix(mark(X)) | → | prefix(X) | | prefix(active(X)) | → | prefix(X) |
Original Signature
Termination of terms over the following signature is verified: app, zWadr, s, prefix, active, mark, from, nil, cons
Strategy
The following SCCs where found
| prefix#(mark(X)) → prefix#(X) | prefix#(active(X)) → prefix#(X) |
| zWadr#(active(X1), X2) → zWadr#(X1, X2) | zWadr#(mark(X1), X2) → zWadr#(X1, X2) |
| zWadr#(X1, active(X2)) → zWadr#(X1, X2) | zWadr#(X1, mark(X2)) → zWadr#(X1, X2) |
| app#(X1, active(X2)) → app#(X1, X2) | app#(X1, mark(X2)) → app#(X1, X2) |
| app#(active(X1), X2) → app#(X1, X2) | app#(mark(X1), X2) → app#(X1, X2) |
| from#(active(X)) → from#(X) | from#(mark(X)) → from#(X) |
| cons#(X1, active(X2)) → cons#(X1, X2) | cons#(mark(X1), X2) → cons#(X1, X2) |
| cons#(X1, mark(X2)) → cons#(X1, X2) | cons#(active(X1), X2) → cons#(X1, X2) |
| s#(mark(X)) → s#(X) | s#(active(X)) → s#(X) |
| mark#(cons(X1, X2)) → active#(cons(mark(X1), X2)) | mark#(s(X)) → active#(s(mark(X))) |
| mark#(zWadr(X1, X2)) → mark#(X2) | mark#(from(X)) → mark#(X) |
| mark#(cons(X1, X2)) → mark#(X1) | active#(from(X)) → mark#(cons(X, from(s(X)))) |
| active#(app(nil, YS)) → mark#(YS) | mark#(prefix(X)) → mark#(X) |
| active#(prefix(L)) → mark#(cons(nil, zWadr(L, prefix(L)))) | mark#(prefix(X)) → active#(prefix(mark(X))) |
| mark#(from(X)) → active#(from(mark(X))) | active#(zWadr(cons(X, XS), cons(Y, YS))) → mark#(cons(app(Y, cons(X, nil)), zWadr(XS, YS))) |
| mark#(app(X1, X2)) → mark#(X1) | mark#(zWadr(X1, X2)) → mark#(X1) |
| mark#(s(X)) → mark#(X) | active#(app(cons(X, XS), YS)) → mark#(cons(X, app(XS, YS))) |
| mark#(app(X1, X2)) → active#(app(mark(X1), mark(X2))) | mark#(zWadr(X1, X2)) → active#(zWadr(mark(X1), mark(X2))) |
| mark#(app(X1, X2)) → mark#(X2) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| app#(X1, active(X2)) | → | app#(X1, X2) | | app#(X1, mark(X2)) | → | app#(X1, X2) |
| app#(active(X1), X2) | → | app#(X1, X2) | | app#(mark(X1), X2) | → | app#(X1, X2) |
Rewrite Rules
| active(app(nil, YS)) | → | mark(YS) | | active(app(cons(X, XS), YS)) | → | mark(cons(X, app(XS, YS))) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(zWadr(nil, YS)) | → | mark(nil) |
| active(zWadr(XS, nil)) | → | mark(nil) | | active(zWadr(cons(X, XS), cons(Y, YS))) | → | mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS))) |
| active(prefix(L)) | → | mark(cons(nil, zWadr(L, prefix(L)))) | | mark(app(X1, X2)) | → | active(app(mark(X1), mark(X2))) |
| mark(nil) | → | active(nil) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(from(X)) | → | active(from(mark(X))) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(zWadr(X1, X2)) | → | active(zWadr(mark(X1), mark(X2))) | | mark(prefix(X)) | → | active(prefix(mark(X))) |
| app(mark(X1), X2) | → | app(X1, X2) | | app(X1, mark(X2)) | → | app(X1, X2) |
| app(active(X1), X2) | → | app(X1, X2) | | app(X1, active(X2)) | → | app(X1, X2) |
| cons(mark(X1), X2) | → | cons(X1, X2) | | cons(X1, mark(X2)) | → | cons(X1, X2) |
| cons(active(X1), X2) | → | cons(X1, X2) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| from(mark(X)) | → | from(X) | | from(active(X)) | → | from(X) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| zWadr(mark(X1), X2) | → | zWadr(X1, X2) | | zWadr(X1, mark(X2)) | → | zWadr(X1, X2) |
| zWadr(active(X1), X2) | → | zWadr(X1, X2) | | zWadr(X1, active(X2)) | → | zWadr(X1, X2) |
| prefix(mark(X)) | → | prefix(X) | | prefix(active(X)) | → | prefix(X) |
Original Signature
Termination of terms over the following signature is verified: app, zWadr, s, prefix, active, mark, from, nil, cons
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| app#(active(X1), X2) | → | app#(X1, X2) | | app#(mark(X1), X2) | → | app#(X1, X2) |
Problem 9: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| app#(X1, active(X2)) | → | app#(X1, X2) | | app#(X1, mark(X2)) | → | app#(X1, X2) |
Rewrite Rules
| active(app(nil, YS)) | → | mark(YS) | | active(app(cons(X, XS), YS)) | → | mark(cons(X, app(XS, YS))) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(zWadr(nil, YS)) | → | mark(nil) |
| active(zWadr(XS, nil)) | → | mark(nil) | | active(zWadr(cons(X, XS), cons(Y, YS))) | → | mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS))) |
| active(prefix(L)) | → | mark(cons(nil, zWadr(L, prefix(L)))) | | mark(app(X1, X2)) | → | active(app(mark(X1), mark(X2))) |
| mark(nil) | → | active(nil) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(from(X)) | → | active(from(mark(X))) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(zWadr(X1, X2)) | → | active(zWadr(mark(X1), mark(X2))) | | mark(prefix(X)) | → | active(prefix(mark(X))) |
| app(mark(X1), X2) | → | app(X1, X2) | | app(X1, mark(X2)) | → | app(X1, X2) |
| app(active(X1), X2) | → | app(X1, X2) | | app(X1, active(X2)) | → | app(X1, X2) |
| cons(mark(X1), X2) | → | cons(X1, X2) | | cons(X1, mark(X2)) | → | cons(X1, X2) |
| cons(active(X1), X2) | → | cons(X1, X2) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| from(mark(X)) | → | from(X) | | from(active(X)) | → | from(X) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| zWadr(mark(X1), X2) | → | zWadr(X1, X2) | | zWadr(X1, mark(X2)) | → | zWadr(X1, X2) |
| zWadr(active(X1), X2) | → | zWadr(X1, X2) | | zWadr(X1, active(X2)) | → | zWadr(X1, X2) |
| prefix(mark(X)) | → | prefix(X) | | prefix(active(X)) | → | prefix(X) |
Original Signature
Termination of terms over the following signature is verified: app, zWadr, s, prefix, active, mark, from, cons, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| app#(X1, active(X2)) | → | app#(X1, X2) | | app#(X1, mark(X2)) | → | app#(X1, X2) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| cons#(X1, active(X2)) | → | cons#(X1, X2) | | cons#(mark(X1), X2) | → | cons#(X1, X2) |
| cons#(X1, mark(X2)) | → | cons#(X1, X2) | | cons#(active(X1), X2) | → | cons#(X1, X2) |
Rewrite Rules
| active(app(nil, YS)) | → | mark(YS) | | active(app(cons(X, XS), YS)) | → | mark(cons(X, app(XS, YS))) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(zWadr(nil, YS)) | → | mark(nil) |
| active(zWadr(XS, nil)) | → | mark(nil) | | active(zWadr(cons(X, XS), cons(Y, YS))) | → | mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS))) |
| active(prefix(L)) | → | mark(cons(nil, zWadr(L, prefix(L)))) | | mark(app(X1, X2)) | → | active(app(mark(X1), mark(X2))) |
| mark(nil) | → | active(nil) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(from(X)) | → | active(from(mark(X))) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(zWadr(X1, X2)) | → | active(zWadr(mark(X1), mark(X2))) | | mark(prefix(X)) | → | active(prefix(mark(X))) |
| app(mark(X1), X2) | → | app(X1, X2) | | app(X1, mark(X2)) | → | app(X1, X2) |
| app(active(X1), X2) | → | app(X1, X2) | | app(X1, active(X2)) | → | app(X1, X2) |
| cons(mark(X1), X2) | → | cons(X1, X2) | | cons(X1, mark(X2)) | → | cons(X1, X2) |
| cons(active(X1), X2) | → | cons(X1, X2) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| from(mark(X)) | → | from(X) | | from(active(X)) | → | from(X) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| zWadr(mark(X1), X2) | → | zWadr(X1, X2) | | zWadr(X1, mark(X2)) | → | zWadr(X1, X2) |
| zWadr(active(X1), X2) | → | zWadr(X1, X2) | | zWadr(X1, active(X2)) | → | zWadr(X1, X2) |
| prefix(mark(X)) | → | prefix(X) | | prefix(active(X)) | → | prefix(X) |
Original Signature
Termination of terms over the following signature is verified: app, zWadr, s, prefix, active, mark, from, nil, cons
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| cons#(mark(X1), X2) | → | cons#(X1, X2) | | cons#(active(X1), X2) | → | cons#(X1, X2) |
Problem 10: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| cons#(X1, active(X2)) | → | cons#(X1, X2) | | cons#(X1, mark(X2)) | → | cons#(X1, X2) |
Rewrite Rules
| active(app(nil, YS)) | → | mark(YS) | | active(app(cons(X, XS), YS)) | → | mark(cons(X, app(XS, YS))) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(zWadr(nil, YS)) | → | mark(nil) |
| active(zWadr(XS, nil)) | → | mark(nil) | | active(zWadr(cons(X, XS), cons(Y, YS))) | → | mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS))) |
| active(prefix(L)) | → | mark(cons(nil, zWadr(L, prefix(L)))) | | mark(app(X1, X2)) | → | active(app(mark(X1), mark(X2))) |
| mark(nil) | → | active(nil) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(from(X)) | → | active(from(mark(X))) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(zWadr(X1, X2)) | → | active(zWadr(mark(X1), mark(X2))) | | mark(prefix(X)) | → | active(prefix(mark(X))) |
| app(mark(X1), X2) | → | app(X1, X2) | | app(X1, mark(X2)) | → | app(X1, X2) |
| app(active(X1), X2) | → | app(X1, X2) | | app(X1, active(X2)) | → | app(X1, X2) |
| cons(mark(X1), X2) | → | cons(X1, X2) | | cons(X1, mark(X2)) | → | cons(X1, X2) |
| cons(active(X1), X2) | → | cons(X1, X2) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| from(mark(X)) | → | from(X) | | from(active(X)) | → | from(X) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| zWadr(mark(X1), X2) | → | zWadr(X1, X2) | | zWadr(X1, mark(X2)) | → | zWadr(X1, X2) |
| zWadr(active(X1), X2) | → | zWadr(X1, X2) | | zWadr(X1, active(X2)) | → | zWadr(X1, X2) |
| prefix(mark(X)) | → | prefix(X) | | prefix(active(X)) | → | prefix(X) |
Original Signature
Termination of terms over the following signature is verified: app, zWadr, s, prefix, active, mark, from, cons, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| cons#(X1, active(X2)) | → | cons#(X1, X2) | | cons#(X1, mark(X2)) | → | cons#(X1, X2) |
Problem 4: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| prefix#(mark(X)) | → | prefix#(X) | | prefix#(active(X)) | → | prefix#(X) |
Rewrite Rules
| active(app(nil, YS)) | → | mark(YS) | | active(app(cons(X, XS), YS)) | → | mark(cons(X, app(XS, YS))) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(zWadr(nil, YS)) | → | mark(nil) |
| active(zWadr(XS, nil)) | → | mark(nil) | | active(zWadr(cons(X, XS), cons(Y, YS))) | → | mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS))) |
| active(prefix(L)) | → | mark(cons(nil, zWadr(L, prefix(L)))) | | mark(app(X1, X2)) | → | active(app(mark(X1), mark(X2))) |
| mark(nil) | → | active(nil) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(from(X)) | → | active(from(mark(X))) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(zWadr(X1, X2)) | → | active(zWadr(mark(X1), mark(X2))) | | mark(prefix(X)) | → | active(prefix(mark(X))) |
| app(mark(X1), X2) | → | app(X1, X2) | | app(X1, mark(X2)) | → | app(X1, X2) |
| app(active(X1), X2) | → | app(X1, X2) | | app(X1, active(X2)) | → | app(X1, X2) |
| cons(mark(X1), X2) | → | cons(X1, X2) | | cons(X1, mark(X2)) | → | cons(X1, X2) |
| cons(active(X1), X2) | → | cons(X1, X2) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| from(mark(X)) | → | from(X) | | from(active(X)) | → | from(X) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| zWadr(mark(X1), X2) | → | zWadr(X1, X2) | | zWadr(X1, mark(X2)) | → | zWadr(X1, X2) |
| zWadr(active(X1), X2) | → | zWadr(X1, X2) | | zWadr(X1, active(X2)) | → | zWadr(X1, X2) |
| prefix(mark(X)) | → | prefix(X) | | prefix(active(X)) | → | prefix(X) |
Original Signature
Termination of terms over the following signature is verified: app, zWadr, s, prefix, active, mark, from, nil, cons
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| prefix#(mark(X)) | → | prefix#(X) | | prefix#(active(X)) | → | prefix#(X) |
Problem 5: PolynomialLinearRange4
Dependency Pair Problem
Dependency Pairs
| mark#(cons(X1, X2)) | → | active#(cons(mark(X1), X2)) | | mark#(s(X)) | → | active#(s(mark(X))) |
| mark#(zWadr(X1, X2)) | → | mark#(X2) | | mark#(from(X)) | → | mark#(X) |
| mark#(cons(X1, X2)) | → | mark#(X1) | | active#(app(nil, YS)) | → | mark#(YS) |
| active#(from(X)) | → | mark#(cons(X, from(s(X)))) | | mark#(prefix(X)) | → | mark#(X) |
| active#(prefix(L)) | → | mark#(cons(nil, zWadr(L, prefix(L)))) | | mark#(prefix(X)) | → | active#(prefix(mark(X))) |
| active#(zWadr(cons(X, XS), cons(Y, YS))) | → | mark#(cons(app(Y, cons(X, nil)), zWadr(XS, YS))) | | mark#(from(X)) | → | active#(from(mark(X))) |
| mark#(app(X1, X2)) | → | mark#(X1) | | mark#(zWadr(X1, X2)) | → | mark#(X1) |
| mark#(s(X)) | → | mark#(X) | | active#(app(cons(X, XS), YS)) | → | mark#(cons(X, app(XS, YS))) |
| mark#(app(X1, X2)) | → | active#(app(mark(X1), mark(X2))) | | mark#(zWadr(X1, X2)) | → | active#(zWadr(mark(X1), mark(X2))) |
| mark#(app(X1, X2)) | → | mark#(X2) |
Rewrite Rules
| active(app(nil, YS)) | → | mark(YS) | | active(app(cons(X, XS), YS)) | → | mark(cons(X, app(XS, YS))) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(zWadr(nil, YS)) | → | mark(nil) |
| active(zWadr(XS, nil)) | → | mark(nil) | | active(zWadr(cons(X, XS), cons(Y, YS))) | → | mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS))) |
| active(prefix(L)) | → | mark(cons(nil, zWadr(L, prefix(L)))) | | mark(app(X1, X2)) | → | active(app(mark(X1), mark(X2))) |
| mark(nil) | → | active(nil) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(from(X)) | → | active(from(mark(X))) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(zWadr(X1, X2)) | → | active(zWadr(mark(X1), mark(X2))) | | mark(prefix(X)) | → | active(prefix(mark(X))) |
| app(mark(X1), X2) | → | app(X1, X2) | | app(X1, mark(X2)) | → | app(X1, X2) |
| app(active(X1), X2) | → | app(X1, X2) | | app(X1, active(X2)) | → | app(X1, X2) |
| cons(mark(X1), X2) | → | cons(X1, X2) | | cons(X1, mark(X2)) | → | cons(X1, X2) |
| cons(active(X1), X2) | → | cons(X1, X2) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| from(mark(X)) | → | from(X) | | from(active(X)) | → | from(X) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| zWadr(mark(X1), X2) | → | zWadr(X1, X2) | | zWadr(X1, mark(X2)) | → | zWadr(X1, X2) |
| zWadr(active(X1), X2) | → | zWadr(X1, X2) | | zWadr(X1, active(X2)) | → | zWadr(X1, X2) |
| prefix(mark(X)) | → | prefix(X) | | prefix(active(X)) | → | prefix(X) |
Original Signature
Termination of terms over the following signature is verified: app, zWadr, s, prefix, active, mark, from, nil, cons
Strategy
Polynomial Interpretation
- active(x): 2x + 1
- active#(x): 2x
- app(x,y): 1
- cons(x,y): 0
- from(x): 1
- mark(x): 3
- mark#(x): 2
- nil: 1
- prefix(x): 1
- s(x): 1
- zWadr(x,y): 1
Standard Usable rules
| cons(active(X1), X2) | → | cons(X1, X2) | | from(mark(X)) | → | from(X) |
| zWadr(X1, mark(X2)) | → | zWadr(X1, X2) | | mark(s(X)) | → | active(s(mark(X))) |
| prefix(active(X)) | → | prefix(X) | | zWadr(active(X1), X2) | → | zWadr(X1, X2) |
| cons(X1, mark(X2)) | → | cons(X1, X2) | | app(X1, mark(X2)) | → | app(X1, X2) |
| active(zWadr(nil, YS)) | → | mark(nil) | | active(zWadr(XS, nil)) | → | mark(nil) |
| mark(from(X)) | → | active(from(mark(X))) | | app(X1, active(X2)) | → | app(X1, X2) |
| app(active(X1), X2) | → | app(X1, X2) | | active(from(X)) | → | mark(cons(X, from(s(X)))) |
| mark(zWadr(X1, X2)) | → | active(zWadr(mark(X1), mark(X2))) | | mark(nil) | → | active(nil) |
| s(active(X)) | → | s(X) | | prefix(mark(X)) | → | prefix(X) |
| active(app(nil, YS)) | → | mark(YS) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| from(active(X)) | → | from(X) | | mark(app(X1, X2)) | → | active(app(mark(X1), mark(X2))) |
| zWadr(X1, active(X2)) | → | zWadr(X1, X2) | | zWadr(mark(X1), X2) | → | zWadr(X1, X2) |
| mark(prefix(X)) | → | active(prefix(mark(X))) | | active(app(cons(X, XS), YS)) | → | mark(cons(X, app(XS, YS))) |
| cons(mark(X1), X2) | → | cons(X1, X2) | | active(zWadr(cons(X, XS), cons(Y, YS))) | → | mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS))) |
| active(prefix(L)) | → | mark(cons(nil, zWadr(L, prefix(L)))) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| s(mark(X)) | → | s(X) | | app(mark(X1), X2) | → | app(X1, X2) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| mark#(cons(X1, X2)) | → | active#(cons(mark(X1), X2)) |
Problem 12: PolynomialLinearRange4
Dependency Pair Problem
Dependency Pairs
| mark#(s(X)) | → | active#(s(mark(X))) | | mark#(zWadr(X1, X2)) | → | mark#(X2) |
| mark#(from(X)) | → | mark#(X) | | mark#(cons(X1, X2)) | → | mark#(X1) |
| active#(app(nil, YS)) | → | mark#(YS) | | active#(from(X)) | → | mark#(cons(X, from(s(X)))) |
| mark#(prefix(X)) | → | mark#(X) | | active#(prefix(L)) | → | mark#(cons(nil, zWadr(L, prefix(L)))) |
| mark#(prefix(X)) | → | active#(prefix(mark(X))) | | active#(zWadr(cons(X, XS), cons(Y, YS))) | → | mark#(cons(app(Y, cons(X, nil)), zWadr(XS, YS))) |
| mark#(from(X)) | → | active#(from(mark(X))) | | mark#(app(X1, X2)) | → | mark#(X1) |
| mark#(zWadr(X1, X2)) | → | mark#(X1) | | mark#(s(X)) | → | mark#(X) |
| active#(app(cons(X, XS), YS)) | → | mark#(cons(X, app(XS, YS))) | | mark#(app(X1, X2)) | → | active#(app(mark(X1), mark(X2))) |
| mark#(zWadr(X1, X2)) | → | active#(zWadr(mark(X1), mark(X2))) | | mark#(app(X1, X2)) | → | mark#(X2) |
Rewrite Rules
| active(app(nil, YS)) | → | mark(YS) | | active(app(cons(X, XS), YS)) | → | mark(cons(X, app(XS, YS))) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(zWadr(nil, YS)) | → | mark(nil) |
| active(zWadr(XS, nil)) | → | mark(nil) | | active(zWadr(cons(X, XS), cons(Y, YS))) | → | mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS))) |
| active(prefix(L)) | → | mark(cons(nil, zWadr(L, prefix(L)))) | | mark(app(X1, X2)) | → | active(app(mark(X1), mark(X2))) |
| mark(nil) | → | active(nil) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(from(X)) | → | active(from(mark(X))) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(zWadr(X1, X2)) | → | active(zWadr(mark(X1), mark(X2))) | | mark(prefix(X)) | → | active(prefix(mark(X))) |
| app(mark(X1), X2) | → | app(X1, X2) | | app(X1, mark(X2)) | → | app(X1, X2) |
| app(active(X1), X2) | → | app(X1, X2) | | app(X1, active(X2)) | → | app(X1, X2) |
| cons(mark(X1), X2) | → | cons(X1, X2) | | cons(X1, mark(X2)) | → | cons(X1, X2) |
| cons(active(X1), X2) | → | cons(X1, X2) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| from(mark(X)) | → | from(X) | | from(active(X)) | → | from(X) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| zWadr(mark(X1), X2) | → | zWadr(X1, X2) | | zWadr(X1, mark(X2)) | → | zWadr(X1, X2) |
| zWadr(active(X1), X2) | → | zWadr(X1, X2) | | zWadr(X1, active(X2)) | → | zWadr(X1, X2) |
| prefix(mark(X)) | → | prefix(X) | | prefix(active(X)) | → | prefix(X) |
Original Signature
Termination of terms over the following signature is verified: app, zWadr, s, prefix, active, mark, from, cons, nil
Strategy
Polynomial Interpretation
- active(x): 0
- active#(x): 2x
- app(x,y): 1
- cons(x,y): 1
- from(x): 1
- mark(x): 0
- mark#(x): 2
- nil: 2
- prefix(x): 1
- s(x): 0
- zWadr(x,y): 1
Standard Usable rules
| cons(active(X1), X2) | → | cons(X1, X2) | | from(mark(X)) | → | from(X) |
| zWadr(X1, mark(X2)) | → | zWadr(X1, X2) | | mark(s(X)) | → | active(s(mark(X))) |
| prefix(active(X)) | → | prefix(X) | | zWadr(active(X1), X2) | → | zWadr(X1, X2) |
| cons(X1, mark(X2)) | → | cons(X1, X2) | | app(X1, mark(X2)) | → | app(X1, X2) |
| active(zWadr(nil, YS)) | → | mark(nil) | | active(zWadr(XS, nil)) | → | mark(nil) |
| mark(from(X)) | → | active(from(mark(X))) | | app(X1, active(X2)) | → | app(X1, X2) |
| app(active(X1), X2) | → | app(X1, X2) | | active(from(X)) | → | mark(cons(X, from(s(X)))) |
| mark(zWadr(X1, X2)) | → | active(zWadr(mark(X1), mark(X2))) | | mark(nil) | → | active(nil) |
| s(active(X)) | → | s(X) | | prefix(mark(X)) | → | prefix(X) |
| active(app(nil, YS)) | → | mark(YS) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| from(active(X)) | → | from(X) | | mark(app(X1, X2)) | → | active(app(mark(X1), mark(X2))) |
| zWadr(X1, active(X2)) | → | zWadr(X1, X2) | | zWadr(mark(X1), X2) | → | zWadr(X1, X2) |
| mark(prefix(X)) | → | active(prefix(mark(X))) | | active(app(cons(X, XS), YS)) | → | mark(cons(X, app(XS, YS))) |
| cons(mark(X1), X2) | → | cons(X1, X2) | | active(zWadr(cons(X, XS), cons(Y, YS))) | → | mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS))) |
| active(prefix(L)) | → | mark(cons(nil, zWadr(L, prefix(L)))) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| s(mark(X)) | → | s(X) | | app(mark(X1), X2) | → | app(X1, X2) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| mark#(s(X)) | → | active#(s(mark(X))) |
Problem 13: PolynomialLinearRange4
Dependency Pair Problem
Dependency Pairs
| mark#(zWadr(X1, X2)) | → | mark#(X2) | | mark#(from(X)) | → | mark#(X) |
| mark#(cons(X1, X2)) | → | mark#(X1) | | active#(app(nil, YS)) | → | mark#(YS) |
| active#(from(X)) | → | mark#(cons(X, from(s(X)))) | | mark#(prefix(X)) | → | mark#(X) |
| active#(prefix(L)) | → | mark#(cons(nil, zWadr(L, prefix(L)))) | | mark#(prefix(X)) | → | active#(prefix(mark(X))) |
| active#(zWadr(cons(X, XS), cons(Y, YS))) | → | mark#(cons(app(Y, cons(X, nil)), zWadr(XS, YS))) | | mark#(from(X)) | → | active#(from(mark(X))) |
| mark#(app(X1, X2)) | → | mark#(X1) | | mark#(zWadr(X1, X2)) | → | mark#(X1) |
| mark#(s(X)) | → | mark#(X) | | active#(app(cons(X, XS), YS)) | → | mark#(cons(X, app(XS, YS))) |
| mark#(app(X1, X2)) | → | active#(app(mark(X1), mark(X2))) | | mark#(zWadr(X1, X2)) | → | active#(zWadr(mark(X1), mark(X2))) |
| mark#(app(X1, X2)) | → | mark#(X2) |
Rewrite Rules
| active(app(nil, YS)) | → | mark(YS) | | active(app(cons(X, XS), YS)) | → | mark(cons(X, app(XS, YS))) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(zWadr(nil, YS)) | → | mark(nil) |
| active(zWadr(XS, nil)) | → | mark(nil) | | active(zWadr(cons(X, XS), cons(Y, YS))) | → | mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS))) |
| active(prefix(L)) | → | mark(cons(nil, zWadr(L, prefix(L)))) | | mark(app(X1, X2)) | → | active(app(mark(X1), mark(X2))) |
| mark(nil) | → | active(nil) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(from(X)) | → | active(from(mark(X))) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(zWadr(X1, X2)) | → | active(zWadr(mark(X1), mark(X2))) | | mark(prefix(X)) | → | active(prefix(mark(X))) |
| app(mark(X1), X2) | → | app(X1, X2) | | app(X1, mark(X2)) | → | app(X1, X2) |
| app(active(X1), X2) | → | app(X1, X2) | | app(X1, active(X2)) | → | app(X1, X2) |
| cons(mark(X1), X2) | → | cons(X1, X2) | | cons(X1, mark(X2)) | → | cons(X1, X2) |
| cons(active(X1), X2) | → | cons(X1, X2) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| from(mark(X)) | → | from(X) | | from(active(X)) | → | from(X) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| zWadr(mark(X1), X2) | → | zWadr(X1, X2) | | zWadr(X1, mark(X2)) | → | zWadr(X1, X2) |
| zWadr(active(X1), X2) | → | zWadr(X1, X2) | | zWadr(X1, active(X2)) | → | zWadr(X1, X2) |
| prefix(mark(X)) | → | prefix(X) | | prefix(active(X)) | → | prefix(X) |
Original Signature
Termination of terms over the following signature is verified: app, zWadr, s, prefix, active, mark, from, nil, cons
Strategy
Polynomial Interpretation
- active(x): x
- active#(x): 2x
- app(x,y): y + x
- cons(x,y): x
- from(x): x
- mark(x): x
- mark#(x): 2x
- nil: 1
- prefix(x): 2x + 1
- s(x): 2x
- zWadr(x,y): y + 2x
Standard Usable rules
| cons(active(X1), X2) | → | cons(X1, X2) | | from(mark(X)) | → | from(X) |
| zWadr(X1, mark(X2)) | → | zWadr(X1, X2) | | mark(s(X)) | → | active(s(mark(X))) |
| prefix(active(X)) | → | prefix(X) | | zWadr(active(X1), X2) | → | zWadr(X1, X2) |
| cons(X1, mark(X2)) | → | cons(X1, X2) | | app(X1, mark(X2)) | → | app(X1, X2) |
| active(zWadr(nil, YS)) | → | mark(nil) | | active(zWadr(XS, nil)) | → | mark(nil) |
| mark(from(X)) | → | active(from(mark(X))) | | app(X1, active(X2)) | → | app(X1, X2) |
| app(active(X1), X2) | → | app(X1, X2) | | active(from(X)) | → | mark(cons(X, from(s(X)))) |
| mark(zWadr(X1, X2)) | → | active(zWadr(mark(X1), mark(X2))) | | mark(nil) | → | active(nil) |
| s(active(X)) | → | s(X) | | prefix(mark(X)) | → | prefix(X) |
| active(app(nil, YS)) | → | mark(YS) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| from(active(X)) | → | from(X) | | mark(app(X1, X2)) | → | active(app(mark(X1), mark(X2))) |
| zWadr(X1, active(X2)) | → | zWadr(X1, X2) | | zWadr(mark(X1), X2) | → | zWadr(X1, X2) |
| mark(prefix(X)) | → | active(prefix(mark(X))) | | active(app(cons(X, XS), YS)) | → | mark(cons(X, app(XS, YS))) |
| cons(mark(X1), X2) | → | cons(X1, X2) | | active(zWadr(cons(X, XS), cons(Y, YS))) | → | mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS))) |
| active(prefix(L)) | → | mark(cons(nil, zWadr(L, prefix(L)))) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| s(mark(X)) | → | s(X) | | app(mark(X1), X2) | → | app(X1, X2) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| active#(app(nil, YS)) | → | mark#(YS) | | mark#(prefix(X)) | → | mark#(X) |
Problem 14: PolynomialLinearRange4
Dependency Pair Problem
Dependency Pairs
| mark#(zWadr(X1, X2)) | → | mark#(X2) | | mark#(from(X)) | → | mark#(X) |
| mark#(cons(X1, X2)) | → | mark#(X1) | | active#(from(X)) | → | mark#(cons(X, from(s(X)))) |
| active#(prefix(L)) | → | mark#(cons(nil, zWadr(L, prefix(L)))) | | mark#(prefix(X)) | → | active#(prefix(mark(X))) |
| active#(zWadr(cons(X, XS), cons(Y, YS))) | → | mark#(cons(app(Y, cons(X, nil)), zWadr(XS, YS))) | | mark#(from(X)) | → | active#(from(mark(X))) |
| mark#(app(X1, X2)) | → | mark#(X1) | | mark#(zWadr(X1, X2)) | → | mark#(X1) |
| mark#(s(X)) | → | mark#(X) | | active#(app(cons(X, XS), YS)) | → | mark#(cons(X, app(XS, YS))) |
| mark#(app(X1, X2)) | → | active#(app(mark(X1), mark(X2))) | | mark#(zWadr(X1, X2)) | → | active#(zWadr(mark(X1), mark(X2))) |
| mark#(app(X1, X2)) | → | mark#(X2) |
Rewrite Rules
| active(app(nil, YS)) | → | mark(YS) | | active(app(cons(X, XS), YS)) | → | mark(cons(X, app(XS, YS))) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(zWadr(nil, YS)) | → | mark(nil) |
| active(zWadr(XS, nil)) | → | mark(nil) | | active(zWadr(cons(X, XS), cons(Y, YS))) | → | mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS))) |
| active(prefix(L)) | → | mark(cons(nil, zWadr(L, prefix(L)))) | | mark(app(X1, X2)) | → | active(app(mark(X1), mark(X2))) |
| mark(nil) | → | active(nil) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(from(X)) | → | active(from(mark(X))) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(zWadr(X1, X2)) | → | active(zWadr(mark(X1), mark(X2))) | | mark(prefix(X)) | → | active(prefix(mark(X))) |
| app(mark(X1), X2) | → | app(X1, X2) | | app(X1, mark(X2)) | → | app(X1, X2) |
| app(active(X1), X2) | → | app(X1, X2) | | app(X1, active(X2)) | → | app(X1, X2) |
| cons(mark(X1), X2) | → | cons(X1, X2) | | cons(X1, mark(X2)) | → | cons(X1, X2) |
| cons(active(X1), X2) | → | cons(X1, X2) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| from(mark(X)) | → | from(X) | | from(active(X)) | → | from(X) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| zWadr(mark(X1), X2) | → | zWadr(X1, X2) | | zWadr(X1, mark(X2)) | → | zWadr(X1, X2) |
| zWadr(active(X1), X2) | → | zWadr(X1, X2) | | zWadr(X1, active(X2)) | → | zWadr(X1, X2) |
| prefix(mark(X)) | → | prefix(X) | | prefix(active(X)) | → | prefix(X) |
Original Signature
Termination of terms over the following signature is verified: app, zWadr, s, prefix, active, mark, from, cons, nil
Strategy
Polynomial Interpretation
- active(x): x
- active#(x): 2x
- app(x,y): 2y + 2x
- cons(x,y): x
- from(x): x
- mark(x): x
- mark#(x): 2x
- nil: 0
- prefix(x): x + 1
- s(x): x
- zWadr(x,y): 2y + 3x
Standard Usable rules
| cons(active(X1), X2) | → | cons(X1, X2) | | from(mark(X)) | → | from(X) |
| zWadr(X1, mark(X2)) | → | zWadr(X1, X2) | | mark(s(X)) | → | active(s(mark(X))) |
| prefix(active(X)) | → | prefix(X) | | zWadr(active(X1), X2) | → | zWadr(X1, X2) |
| cons(X1, mark(X2)) | → | cons(X1, X2) | | app(X1, mark(X2)) | → | app(X1, X2) |
| active(zWadr(nil, YS)) | → | mark(nil) | | active(zWadr(XS, nil)) | → | mark(nil) |
| mark(from(X)) | → | active(from(mark(X))) | | app(X1, active(X2)) | → | app(X1, X2) |
| app(active(X1), X2) | → | app(X1, X2) | | active(from(X)) | → | mark(cons(X, from(s(X)))) |
| mark(zWadr(X1, X2)) | → | active(zWadr(mark(X1), mark(X2))) | | mark(nil) | → | active(nil) |
| s(active(X)) | → | s(X) | | prefix(mark(X)) | → | prefix(X) |
| active(app(nil, YS)) | → | mark(YS) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| from(active(X)) | → | from(X) | | mark(app(X1, X2)) | → | active(app(mark(X1), mark(X2))) |
| zWadr(X1, active(X2)) | → | zWadr(X1, X2) | | zWadr(mark(X1), X2) | → | zWadr(X1, X2) |
| mark(prefix(X)) | → | active(prefix(mark(X))) | | active(app(cons(X, XS), YS)) | → | mark(cons(X, app(XS, YS))) |
| cons(mark(X1), X2) | → | cons(X1, X2) | | active(zWadr(cons(X, XS), cons(Y, YS))) | → | mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS))) |
| active(prefix(L)) | → | mark(cons(nil, zWadr(L, prefix(L)))) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| s(mark(X)) | → | s(X) | | app(mark(X1), X2) | → | app(X1, X2) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| active#(prefix(L)) | → | mark#(cons(nil, zWadr(L, prefix(L)))) |
Problem 15: PolynomialLinearRange4
Dependency Pair Problem
Dependency Pairs
| mark#(zWadr(X1, X2)) | → | mark#(X2) | | mark#(from(X)) | → | mark#(X) |
| mark#(cons(X1, X2)) | → | mark#(X1) | | active#(from(X)) | → | mark#(cons(X, from(s(X)))) |
| mark#(prefix(X)) | → | active#(prefix(mark(X))) | | active#(zWadr(cons(X, XS), cons(Y, YS))) | → | mark#(cons(app(Y, cons(X, nil)), zWadr(XS, YS))) |
| mark#(from(X)) | → | active#(from(mark(X))) | | mark#(app(X1, X2)) | → | mark#(X1) |
| mark#(zWadr(X1, X2)) | → | mark#(X1) | | mark#(s(X)) | → | mark#(X) |
| active#(app(cons(X, XS), YS)) | → | mark#(cons(X, app(XS, YS))) | | mark#(app(X1, X2)) | → | active#(app(mark(X1), mark(X2))) |
| mark#(zWadr(X1, X2)) | → | active#(zWadr(mark(X1), mark(X2))) | | mark#(app(X1, X2)) | → | mark#(X2) |
Rewrite Rules
| active(app(nil, YS)) | → | mark(YS) | | active(app(cons(X, XS), YS)) | → | mark(cons(X, app(XS, YS))) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(zWadr(nil, YS)) | → | mark(nil) |
| active(zWadr(XS, nil)) | → | mark(nil) | | active(zWadr(cons(X, XS), cons(Y, YS))) | → | mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS))) |
| active(prefix(L)) | → | mark(cons(nil, zWadr(L, prefix(L)))) | | mark(app(X1, X2)) | → | active(app(mark(X1), mark(X2))) |
| mark(nil) | → | active(nil) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(from(X)) | → | active(from(mark(X))) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(zWadr(X1, X2)) | → | active(zWadr(mark(X1), mark(X2))) | | mark(prefix(X)) | → | active(prefix(mark(X))) |
| app(mark(X1), X2) | → | app(X1, X2) | | app(X1, mark(X2)) | → | app(X1, X2) |
| app(active(X1), X2) | → | app(X1, X2) | | app(X1, active(X2)) | → | app(X1, X2) |
| cons(mark(X1), X2) | → | cons(X1, X2) | | cons(X1, mark(X2)) | → | cons(X1, X2) |
| cons(active(X1), X2) | → | cons(X1, X2) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| from(mark(X)) | → | from(X) | | from(active(X)) | → | from(X) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| zWadr(mark(X1), X2) | → | zWadr(X1, X2) | | zWadr(X1, mark(X2)) | → | zWadr(X1, X2) |
| zWadr(active(X1), X2) | → | zWadr(X1, X2) | | zWadr(X1, active(X2)) | → | zWadr(X1, X2) |
| prefix(mark(X)) | → | prefix(X) | | prefix(active(X)) | → | prefix(X) |
Original Signature
Termination of terms over the following signature is verified: app, zWadr, s, prefix, active, mark, from, nil, cons
Strategy
Polynomial Interpretation
- active(x): 1
- active#(x): x
- app(x,y): 2
- cons(x,y): 0
- from(x): 2
- mark(x): 1
- mark#(x): 2
- nil: 3
- prefix(x): 0
- s(x): 0
- zWadr(x,y): 2
Standard Usable rules
| cons(active(X1), X2) | → | cons(X1, X2) | | from(mark(X)) | → | from(X) |
| zWadr(X1, mark(X2)) | → | zWadr(X1, X2) | | mark(s(X)) | → | active(s(mark(X))) |
| prefix(active(X)) | → | prefix(X) | | zWadr(active(X1), X2) | → | zWadr(X1, X2) |
| cons(X1, mark(X2)) | → | cons(X1, X2) | | app(X1, mark(X2)) | → | app(X1, X2) |
| active(zWadr(nil, YS)) | → | mark(nil) | | active(zWadr(XS, nil)) | → | mark(nil) |
| mark(from(X)) | → | active(from(mark(X))) | | app(X1, active(X2)) | → | app(X1, X2) |
| app(active(X1), X2) | → | app(X1, X2) | | active(from(X)) | → | mark(cons(X, from(s(X)))) |
| mark(zWadr(X1, X2)) | → | active(zWadr(mark(X1), mark(X2))) | | mark(nil) | → | active(nil) |
| s(active(X)) | → | s(X) | | prefix(mark(X)) | → | prefix(X) |
| active(app(nil, YS)) | → | mark(YS) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| from(active(X)) | → | from(X) | | mark(app(X1, X2)) | → | active(app(mark(X1), mark(X2))) |
| zWadr(X1, active(X2)) | → | zWadr(X1, X2) | | zWadr(mark(X1), X2) | → | zWadr(X1, X2) |
| mark(prefix(X)) | → | active(prefix(mark(X))) | | active(app(cons(X, XS), YS)) | → | mark(cons(X, app(XS, YS))) |
| cons(mark(X1), X2) | → | cons(X1, X2) | | active(zWadr(cons(X, XS), cons(Y, YS))) | → | mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS))) |
| active(prefix(L)) | → | mark(cons(nil, zWadr(L, prefix(L)))) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| s(mark(X)) | → | s(X) | | app(mark(X1), X2) | → | app(X1, X2) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| mark#(prefix(X)) | → | active#(prefix(mark(X))) |
Problem 16: PolynomialLinearRange4
Dependency Pair Problem
Dependency Pairs
| mark#(zWadr(X1, X2)) | → | mark#(X2) | | mark#(from(X)) | → | mark#(X) |
| mark#(cons(X1, X2)) | → | mark#(X1) | | active#(from(X)) | → | mark#(cons(X, from(s(X)))) |
| active#(zWadr(cons(X, XS), cons(Y, YS))) | → | mark#(cons(app(Y, cons(X, nil)), zWadr(XS, YS))) | | mark#(from(X)) | → | active#(from(mark(X))) |
| mark#(app(X1, X2)) | → | mark#(X1) | | mark#(zWadr(X1, X2)) | → | mark#(X1) |
| mark#(s(X)) | → | mark#(X) | | active#(app(cons(X, XS), YS)) | → | mark#(cons(X, app(XS, YS))) |
| mark#(app(X1, X2)) | → | active#(app(mark(X1), mark(X2))) | | mark#(zWadr(X1, X2)) | → | active#(zWadr(mark(X1), mark(X2))) |
| mark#(app(X1, X2)) | → | mark#(X2) |
Rewrite Rules
| active(app(nil, YS)) | → | mark(YS) | | active(app(cons(X, XS), YS)) | → | mark(cons(X, app(XS, YS))) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(zWadr(nil, YS)) | → | mark(nil) |
| active(zWadr(XS, nil)) | → | mark(nil) | | active(zWadr(cons(X, XS), cons(Y, YS))) | → | mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS))) |
| active(prefix(L)) | → | mark(cons(nil, zWadr(L, prefix(L)))) | | mark(app(X1, X2)) | → | active(app(mark(X1), mark(X2))) |
| mark(nil) | → | active(nil) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(from(X)) | → | active(from(mark(X))) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(zWadr(X1, X2)) | → | active(zWadr(mark(X1), mark(X2))) | | mark(prefix(X)) | → | active(prefix(mark(X))) |
| app(mark(X1), X2) | → | app(X1, X2) | | app(X1, mark(X2)) | → | app(X1, X2) |
| app(active(X1), X2) | → | app(X1, X2) | | app(X1, active(X2)) | → | app(X1, X2) |
| cons(mark(X1), X2) | → | cons(X1, X2) | | cons(X1, mark(X2)) | → | cons(X1, X2) |
| cons(active(X1), X2) | → | cons(X1, X2) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| from(mark(X)) | → | from(X) | | from(active(X)) | → | from(X) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| zWadr(mark(X1), X2) | → | zWadr(X1, X2) | | zWadr(X1, mark(X2)) | → | zWadr(X1, X2) |
| zWadr(active(X1), X2) | → | zWadr(X1, X2) | | zWadr(X1, active(X2)) | → | zWadr(X1, X2) |
| prefix(mark(X)) | → | prefix(X) | | prefix(active(X)) | → | prefix(X) |
Original Signature
Termination of terms over the following signature is verified: app, zWadr, s, prefix, active, mark, from, cons, nil
Strategy
Polynomial Interpretation
- active(x): x
- active#(x): 2x + 1
- app(x,y): y + 2x + 1
- cons(x,y): x
- from(x): 2x + 1
- mark(x): x
- mark#(x): 3x
- nil: 0
- prefix(x): 0
- s(x): x
- zWadr(x,y): 3y + 2x + 1
Standard Usable rules
| cons(active(X1), X2) | → | cons(X1, X2) | | from(mark(X)) | → | from(X) |
| zWadr(X1, mark(X2)) | → | zWadr(X1, X2) | | mark(s(X)) | → | active(s(mark(X))) |
| prefix(active(X)) | → | prefix(X) | | zWadr(active(X1), X2) | → | zWadr(X1, X2) |
| cons(X1, mark(X2)) | → | cons(X1, X2) | | app(X1, mark(X2)) | → | app(X1, X2) |
| active(zWadr(XS, nil)) | → | mark(nil) | | active(zWadr(nil, YS)) | → | mark(nil) |
| mark(from(X)) | → | active(from(mark(X))) | | app(X1, active(X2)) | → | app(X1, X2) |
| app(active(X1), X2) | → | app(X1, X2) | | active(from(X)) | → | mark(cons(X, from(s(X)))) |
| mark(zWadr(X1, X2)) | → | active(zWadr(mark(X1), mark(X2))) | | mark(nil) | → | active(nil) |
| prefix(mark(X)) | → | prefix(X) | | s(active(X)) | → | s(X) |
| active(app(nil, YS)) | → | mark(YS) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| from(active(X)) | → | from(X) | | mark(app(X1, X2)) | → | active(app(mark(X1), mark(X2))) |
| zWadr(X1, active(X2)) | → | zWadr(X1, X2) | | zWadr(mark(X1), X2) | → | zWadr(X1, X2) |
| mark(prefix(X)) | → | active(prefix(mark(X))) | | active(app(cons(X, XS), YS)) | → | mark(cons(X, app(XS, YS))) |
| cons(mark(X1), X2) | → | cons(X1, X2) | | active(zWadr(cons(X, XS), cons(Y, YS))) | → | mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS))) |
| active(prefix(L)) | → | mark(cons(nil, zWadr(L, prefix(L)))) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| s(mark(X)) | → | s(X) | | app(mark(X1), X2) | → | app(X1, X2) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| mark#(zWadr(X1, X2)) | → | mark#(X2) | | mark#(from(X)) | → | mark#(X) |
| active#(from(X)) | → | mark#(cons(X, from(s(X)))) | | mark#(app(X1, X2)) | → | mark#(X1) |
| mark#(zWadr(X1, X2)) | → | mark#(X1) | | active#(app(cons(X, XS), YS)) | → | mark#(cons(X, app(XS, YS))) |
| mark#(app(X1, X2)) | → | mark#(X2) |
Problem 17: PolynomialLinearRange4
Dependency Pair Problem
Dependency Pairs
| mark#(from(X)) | → | active#(from(mark(X))) | | active#(zWadr(cons(X, XS), cons(Y, YS))) | → | mark#(cons(app(Y, cons(X, nil)), zWadr(XS, YS))) |
| mark#(cons(X1, X2)) | → | mark#(X1) | | mark#(s(X)) | → | mark#(X) |
| mark#(app(X1, X2)) | → | active#(app(mark(X1), mark(X2))) | | mark#(zWadr(X1, X2)) | → | active#(zWadr(mark(X1), mark(X2))) |
Rewrite Rules
| active(app(nil, YS)) | → | mark(YS) | | active(app(cons(X, XS), YS)) | → | mark(cons(X, app(XS, YS))) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(zWadr(nil, YS)) | → | mark(nil) |
| active(zWadr(XS, nil)) | → | mark(nil) | | active(zWadr(cons(X, XS), cons(Y, YS))) | → | mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS))) |
| active(prefix(L)) | → | mark(cons(nil, zWadr(L, prefix(L)))) | | mark(app(X1, X2)) | → | active(app(mark(X1), mark(X2))) |
| mark(nil) | → | active(nil) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(from(X)) | → | active(from(mark(X))) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(zWadr(X1, X2)) | → | active(zWadr(mark(X1), mark(X2))) | | mark(prefix(X)) | → | active(prefix(mark(X))) |
| app(mark(X1), X2) | → | app(X1, X2) | | app(X1, mark(X2)) | → | app(X1, X2) |
| app(active(X1), X2) | → | app(X1, X2) | | app(X1, active(X2)) | → | app(X1, X2) |
| cons(mark(X1), X2) | → | cons(X1, X2) | | cons(X1, mark(X2)) | → | cons(X1, X2) |
| cons(active(X1), X2) | → | cons(X1, X2) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| from(mark(X)) | → | from(X) | | from(active(X)) | → | from(X) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| zWadr(mark(X1), X2) | → | zWadr(X1, X2) | | zWadr(X1, mark(X2)) | → | zWadr(X1, X2) |
| zWadr(active(X1), X2) | → | zWadr(X1, X2) | | zWadr(X1, active(X2)) | → | zWadr(X1, X2) |
| prefix(mark(X)) | → | prefix(X) | | prefix(active(X)) | → | prefix(X) |
Original Signature
Termination of terms over the following signature is verified: app, zWadr, s, prefix, active, mark, from, nil, cons
Strategy
Polynomial Interpretation
- active(x): 1
- active#(x): x
- app(x,y): 0
- cons(x,y): 1
- from(x): 1
- mark(x): 1
- mark#(x): 1
- nil: 3
- prefix(x): 1
- s(x): 1
- zWadr(x,y): 1
Standard Usable rules
| cons(active(X1), X2) | → | cons(X1, X2) | | from(mark(X)) | → | from(X) |
| zWadr(X1, mark(X2)) | → | zWadr(X1, X2) | | mark(s(X)) | → | active(s(mark(X))) |
| prefix(active(X)) | → | prefix(X) | | zWadr(active(X1), X2) | → | zWadr(X1, X2) |
| cons(X1, mark(X2)) | → | cons(X1, X2) | | app(X1, mark(X2)) | → | app(X1, X2) |
| active(zWadr(XS, nil)) | → | mark(nil) | | active(zWadr(nil, YS)) | → | mark(nil) |
| mark(from(X)) | → | active(from(mark(X))) | | app(X1, active(X2)) | → | app(X1, X2) |
| app(active(X1), X2) | → | app(X1, X2) | | active(from(X)) | → | mark(cons(X, from(s(X)))) |
| mark(zWadr(X1, X2)) | → | active(zWadr(mark(X1), mark(X2))) | | mark(nil) | → | active(nil) |
| s(active(X)) | → | s(X) | | prefix(mark(X)) | → | prefix(X) |
| active(app(nil, YS)) | → | mark(YS) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| from(active(X)) | → | from(X) | | mark(app(X1, X2)) | → | active(app(mark(X1), mark(X2))) |
| zWadr(X1, active(X2)) | → | zWadr(X1, X2) | | zWadr(mark(X1), X2) | → | zWadr(X1, X2) |
| mark(prefix(X)) | → | active(prefix(mark(X))) | | cons(mark(X1), X2) | → | cons(X1, X2) |
| active(app(cons(X, XS), YS)) | → | mark(cons(X, app(XS, YS))) | | active(zWadr(cons(X, XS), cons(Y, YS))) | → | mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS))) |
| mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) | | active(prefix(L)) | → | mark(cons(nil, zWadr(L, prefix(L)))) |
| s(mark(X)) | → | s(X) | | app(mark(X1), X2) | → | app(X1, X2) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| mark#(app(X1, X2)) | → | active#(app(mark(X1), mark(X2))) |
Problem 18: PolynomialLinearRange4
Dependency Pair Problem
Dependency Pairs
| active#(zWadr(cons(X, XS), cons(Y, YS))) | → | mark#(cons(app(Y, cons(X, nil)), zWadr(XS, YS))) | | mark#(from(X)) | → | active#(from(mark(X))) |
| mark#(cons(X1, X2)) | → | mark#(X1) | | mark#(s(X)) | → | mark#(X) |
| mark#(zWadr(X1, X2)) | → | active#(zWadr(mark(X1), mark(X2))) |
Rewrite Rules
| active(app(nil, YS)) | → | mark(YS) | | active(app(cons(X, XS), YS)) | → | mark(cons(X, app(XS, YS))) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(zWadr(nil, YS)) | → | mark(nil) |
| active(zWadr(XS, nil)) | → | mark(nil) | | active(zWadr(cons(X, XS), cons(Y, YS))) | → | mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS))) |
| active(prefix(L)) | → | mark(cons(nil, zWadr(L, prefix(L)))) | | mark(app(X1, X2)) | → | active(app(mark(X1), mark(X2))) |
| mark(nil) | → | active(nil) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(from(X)) | → | active(from(mark(X))) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(zWadr(X1, X2)) | → | active(zWadr(mark(X1), mark(X2))) | | mark(prefix(X)) | → | active(prefix(mark(X))) |
| app(mark(X1), X2) | → | app(X1, X2) | | app(X1, mark(X2)) | → | app(X1, X2) |
| app(active(X1), X2) | → | app(X1, X2) | | app(X1, active(X2)) | → | app(X1, X2) |
| cons(mark(X1), X2) | → | cons(X1, X2) | | cons(X1, mark(X2)) | → | cons(X1, X2) |
| cons(active(X1), X2) | → | cons(X1, X2) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| from(mark(X)) | → | from(X) | | from(active(X)) | → | from(X) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| zWadr(mark(X1), X2) | → | zWadr(X1, X2) | | zWadr(X1, mark(X2)) | → | zWadr(X1, X2) |
| zWadr(active(X1), X2) | → | zWadr(X1, X2) | | zWadr(X1, active(X2)) | → | zWadr(X1, X2) |
| prefix(mark(X)) | → | prefix(X) | | prefix(active(X)) | → | prefix(X) |
Original Signature
Termination of terms over the following signature is verified: app, zWadr, s, prefix, active, mark, from, cons, nil
Strategy
Polynomial Interpretation
- active(x): x
- active#(x): 3x
- app(x,y): y + x
- cons(x,y): x
- from(x): 3x
- mark(x): x
- mark#(x): 3x
- nil: 1
- prefix(x): 1
- s(x): 3x
- zWadr(x,y): 2y + x + 1
Standard Usable rules
| cons(active(X1), X2) | → | cons(X1, X2) | | from(mark(X)) | → | from(X) |
| zWadr(X1, mark(X2)) | → | zWadr(X1, X2) | | mark(s(X)) | → | active(s(mark(X))) |
| prefix(active(X)) | → | prefix(X) | | zWadr(active(X1), X2) | → | zWadr(X1, X2) |
| cons(X1, mark(X2)) | → | cons(X1, X2) | | app(X1, mark(X2)) | → | app(X1, X2) |
| active(zWadr(XS, nil)) | → | mark(nil) | | active(zWadr(nil, YS)) | → | mark(nil) |
| mark(from(X)) | → | active(from(mark(X))) | | app(X1, active(X2)) | → | app(X1, X2) |
| app(active(X1), X2) | → | app(X1, X2) | | active(from(X)) | → | mark(cons(X, from(s(X)))) |
| mark(zWadr(X1, X2)) | → | active(zWadr(mark(X1), mark(X2))) | | mark(nil) | → | active(nil) |
| s(active(X)) | → | s(X) | | prefix(mark(X)) | → | prefix(X) |
| active(app(nil, YS)) | → | mark(YS) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| from(active(X)) | → | from(X) | | mark(app(X1, X2)) | → | active(app(mark(X1), mark(X2))) |
| zWadr(X1, active(X2)) | → | zWadr(X1, X2) | | zWadr(mark(X1), X2) | → | zWadr(X1, X2) |
| mark(prefix(X)) | → | active(prefix(mark(X))) | | cons(mark(X1), X2) | → | cons(X1, X2) |
| active(app(cons(X, XS), YS)) | → | mark(cons(X, app(XS, YS))) | | active(zWadr(cons(X, XS), cons(Y, YS))) | → | mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS))) |
| mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) | | active(prefix(L)) | → | mark(cons(nil, zWadr(L, prefix(L)))) |
| s(mark(X)) | → | s(X) | | app(mark(X1), X2) | → | app(X1, X2) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| active#(zWadr(cons(X, XS), cons(Y, YS))) | → | mark#(cons(app(Y, cons(X, nil)), zWadr(XS, YS))) |
Problem 19: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| mark#(from(X)) | → | active#(from(mark(X))) | | mark#(cons(X1, X2)) | → | mark#(X1) |
| mark#(s(X)) | → | mark#(X) | | mark#(zWadr(X1, X2)) | → | active#(zWadr(mark(X1), mark(X2))) |
Rewrite Rules
| active(app(nil, YS)) | → | mark(YS) | | active(app(cons(X, XS), YS)) | → | mark(cons(X, app(XS, YS))) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(zWadr(nil, YS)) | → | mark(nil) |
| active(zWadr(XS, nil)) | → | mark(nil) | | active(zWadr(cons(X, XS), cons(Y, YS))) | → | mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS))) |
| active(prefix(L)) | → | mark(cons(nil, zWadr(L, prefix(L)))) | | mark(app(X1, X2)) | → | active(app(mark(X1), mark(X2))) |
| mark(nil) | → | active(nil) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(from(X)) | → | active(from(mark(X))) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(zWadr(X1, X2)) | → | active(zWadr(mark(X1), mark(X2))) | | mark(prefix(X)) | → | active(prefix(mark(X))) |
| app(mark(X1), X2) | → | app(X1, X2) | | app(X1, mark(X2)) | → | app(X1, X2) |
| app(active(X1), X2) | → | app(X1, X2) | | app(X1, active(X2)) | → | app(X1, X2) |
| cons(mark(X1), X2) | → | cons(X1, X2) | | cons(X1, mark(X2)) | → | cons(X1, X2) |
| cons(active(X1), X2) | → | cons(X1, X2) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| from(mark(X)) | → | from(X) | | from(active(X)) | → | from(X) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| zWadr(mark(X1), X2) | → | zWadr(X1, X2) | | zWadr(X1, mark(X2)) | → | zWadr(X1, X2) |
| zWadr(active(X1), X2) | → | zWadr(X1, X2) | | zWadr(X1, active(X2)) | → | zWadr(X1, X2) |
| prefix(mark(X)) | → | prefix(X) | | prefix(active(X)) | → | prefix(X) |
Original Signature
Termination of terms over the following signature is verified: app, zWadr, s, prefix, active, mark, from, nil, cons
Strategy
The following SCCs where found
| mark#(cons(X1, X2)) → mark#(X1) | mark#(s(X)) → mark#(X) |
Problem 20: PolynomialLinearRange4
Dependency Pair Problem
Dependency Pairs
| mark#(cons(X1, X2)) | → | mark#(X1) | | mark#(s(X)) | → | mark#(X) |
Rewrite Rules
| active(app(nil, YS)) | → | mark(YS) | | active(app(cons(X, XS), YS)) | → | mark(cons(X, app(XS, YS))) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(zWadr(nil, YS)) | → | mark(nil) |
| active(zWadr(XS, nil)) | → | mark(nil) | | active(zWadr(cons(X, XS), cons(Y, YS))) | → | mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS))) |
| active(prefix(L)) | → | mark(cons(nil, zWadr(L, prefix(L)))) | | mark(app(X1, X2)) | → | active(app(mark(X1), mark(X2))) |
| mark(nil) | → | active(nil) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(from(X)) | → | active(from(mark(X))) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(zWadr(X1, X2)) | → | active(zWadr(mark(X1), mark(X2))) | | mark(prefix(X)) | → | active(prefix(mark(X))) |
| app(mark(X1), X2) | → | app(X1, X2) | | app(X1, mark(X2)) | → | app(X1, X2) |
| app(active(X1), X2) | → | app(X1, X2) | | app(X1, active(X2)) | → | app(X1, X2) |
| cons(mark(X1), X2) | → | cons(X1, X2) | | cons(X1, mark(X2)) | → | cons(X1, X2) |
| cons(active(X1), X2) | → | cons(X1, X2) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| from(mark(X)) | → | from(X) | | from(active(X)) | → | from(X) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| zWadr(mark(X1), X2) | → | zWadr(X1, X2) | | zWadr(X1, mark(X2)) | → | zWadr(X1, X2) |
| zWadr(active(X1), X2) | → | zWadr(X1, X2) | | zWadr(X1, active(X2)) | → | zWadr(X1, X2) |
| prefix(mark(X)) | → | prefix(X) | | prefix(active(X)) | → | prefix(X) |
Original Signature
Termination of terms over the following signature is verified: app, zWadr, s, prefix, active, mark, from, nil, cons
Strategy
Polynomial Interpretation
- active(x): 0
- app(x,y): 0
- cons(x,y): x + 2
- from(x): 0
- mark(x): 0
- mark#(x): x
- nil: 0
- prefix(x): 0
- s(x): x
- zWadr(x,y): 0
There are no usable rules
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| mark#(cons(X1, X2)) | → | mark#(X1) |
Problem 21: PolynomialLinearRange4
Dependency Pair Problem
Dependency Pairs
Rewrite Rules
| active(app(nil, YS)) | → | mark(YS) | | active(app(cons(X, XS), YS)) | → | mark(cons(X, app(XS, YS))) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(zWadr(nil, YS)) | → | mark(nil) |
| active(zWadr(XS, nil)) | → | mark(nil) | | active(zWadr(cons(X, XS), cons(Y, YS))) | → | mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS))) |
| active(prefix(L)) | → | mark(cons(nil, zWadr(L, prefix(L)))) | | mark(app(X1, X2)) | → | active(app(mark(X1), mark(X2))) |
| mark(nil) | → | active(nil) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(from(X)) | → | active(from(mark(X))) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(zWadr(X1, X2)) | → | active(zWadr(mark(X1), mark(X2))) | | mark(prefix(X)) | → | active(prefix(mark(X))) |
| app(mark(X1), X2) | → | app(X1, X2) | | app(X1, mark(X2)) | → | app(X1, X2) |
| app(active(X1), X2) | → | app(X1, X2) | | app(X1, active(X2)) | → | app(X1, X2) |
| cons(mark(X1), X2) | → | cons(X1, X2) | | cons(X1, mark(X2)) | → | cons(X1, X2) |
| cons(active(X1), X2) | → | cons(X1, X2) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| from(mark(X)) | → | from(X) | | from(active(X)) | → | from(X) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| zWadr(mark(X1), X2) | → | zWadr(X1, X2) | | zWadr(X1, mark(X2)) | → | zWadr(X1, X2) |
| zWadr(active(X1), X2) | → | zWadr(X1, X2) | | zWadr(X1, active(X2)) | → | zWadr(X1, X2) |
| prefix(mark(X)) | → | prefix(X) | | prefix(active(X)) | → | prefix(X) |
Original Signature
Termination of terms over the following signature is verified: app, zWadr, s, prefix, active, mark, from, cons, nil
Strategy
Polynomial Interpretation
- active(x): 0
- app(x,y): 0
- cons(x,y): 0
- from(x): 0
- mark(x): 0
- mark#(x): x
- nil: 0
- prefix(x): 0
- s(x): x + 2
- zWadr(x,y): 0
There are no usable rules
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
Problem 6: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| from#(active(X)) | → | from#(X) | | from#(mark(X)) | → | from#(X) |
Rewrite Rules
| active(app(nil, YS)) | → | mark(YS) | | active(app(cons(X, XS), YS)) | → | mark(cons(X, app(XS, YS))) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(zWadr(nil, YS)) | → | mark(nil) |
| active(zWadr(XS, nil)) | → | mark(nil) | | active(zWadr(cons(X, XS), cons(Y, YS))) | → | mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS))) |
| active(prefix(L)) | → | mark(cons(nil, zWadr(L, prefix(L)))) | | mark(app(X1, X2)) | → | active(app(mark(X1), mark(X2))) |
| mark(nil) | → | active(nil) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(from(X)) | → | active(from(mark(X))) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(zWadr(X1, X2)) | → | active(zWadr(mark(X1), mark(X2))) | | mark(prefix(X)) | → | active(prefix(mark(X))) |
| app(mark(X1), X2) | → | app(X1, X2) | | app(X1, mark(X2)) | → | app(X1, X2) |
| app(active(X1), X2) | → | app(X1, X2) | | app(X1, active(X2)) | → | app(X1, X2) |
| cons(mark(X1), X2) | → | cons(X1, X2) | | cons(X1, mark(X2)) | → | cons(X1, X2) |
| cons(active(X1), X2) | → | cons(X1, X2) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| from(mark(X)) | → | from(X) | | from(active(X)) | → | from(X) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| zWadr(mark(X1), X2) | → | zWadr(X1, X2) | | zWadr(X1, mark(X2)) | → | zWadr(X1, X2) |
| zWadr(active(X1), X2) | → | zWadr(X1, X2) | | zWadr(X1, active(X2)) | → | zWadr(X1, X2) |
| prefix(mark(X)) | → | prefix(X) | | prefix(active(X)) | → | prefix(X) |
Original Signature
Termination of terms over the following signature is verified: app, zWadr, s, prefix, active, mark, from, nil, cons
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| from#(active(X)) | → | from#(X) | | from#(mark(X)) | → | from#(X) |
Problem 7: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| s#(mark(X)) | → | s#(X) | | s#(active(X)) | → | s#(X) |
Rewrite Rules
| active(app(nil, YS)) | → | mark(YS) | | active(app(cons(X, XS), YS)) | → | mark(cons(X, app(XS, YS))) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(zWadr(nil, YS)) | → | mark(nil) |
| active(zWadr(XS, nil)) | → | mark(nil) | | active(zWadr(cons(X, XS), cons(Y, YS))) | → | mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS))) |
| active(prefix(L)) | → | mark(cons(nil, zWadr(L, prefix(L)))) | | mark(app(X1, X2)) | → | active(app(mark(X1), mark(X2))) |
| mark(nil) | → | active(nil) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(from(X)) | → | active(from(mark(X))) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(zWadr(X1, X2)) | → | active(zWadr(mark(X1), mark(X2))) | | mark(prefix(X)) | → | active(prefix(mark(X))) |
| app(mark(X1), X2) | → | app(X1, X2) | | app(X1, mark(X2)) | → | app(X1, X2) |
| app(active(X1), X2) | → | app(X1, X2) | | app(X1, active(X2)) | → | app(X1, X2) |
| cons(mark(X1), X2) | → | cons(X1, X2) | | cons(X1, mark(X2)) | → | cons(X1, X2) |
| cons(active(X1), X2) | → | cons(X1, X2) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| from(mark(X)) | → | from(X) | | from(active(X)) | → | from(X) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| zWadr(mark(X1), X2) | → | zWadr(X1, X2) | | zWadr(X1, mark(X2)) | → | zWadr(X1, X2) |
| zWadr(active(X1), X2) | → | zWadr(X1, X2) | | zWadr(X1, active(X2)) | → | zWadr(X1, X2) |
| prefix(mark(X)) | → | prefix(X) | | prefix(active(X)) | → | prefix(X) |
Original Signature
Termination of terms over the following signature is verified: app, zWadr, s, prefix, active, mark, from, nil, cons
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| s#(mark(X)) | → | s#(X) | | s#(active(X)) | → | s#(X) |
Problem 8: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| zWadr#(active(X1), X2) | → | zWadr#(X1, X2) | | zWadr#(mark(X1), X2) | → | zWadr#(X1, X2) |
| zWadr#(X1, active(X2)) | → | zWadr#(X1, X2) | | zWadr#(X1, mark(X2)) | → | zWadr#(X1, X2) |
Rewrite Rules
| active(app(nil, YS)) | → | mark(YS) | | active(app(cons(X, XS), YS)) | → | mark(cons(X, app(XS, YS))) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(zWadr(nil, YS)) | → | mark(nil) |
| active(zWadr(XS, nil)) | → | mark(nil) | | active(zWadr(cons(X, XS), cons(Y, YS))) | → | mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS))) |
| active(prefix(L)) | → | mark(cons(nil, zWadr(L, prefix(L)))) | | mark(app(X1, X2)) | → | active(app(mark(X1), mark(X2))) |
| mark(nil) | → | active(nil) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(from(X)) | → | active(from(mark(X))) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(zWadr(X1, X2)) | → | active(zWadr(mark(X1), mark(X2))) | | mark(prefix(X)) | → | active(prefix(mark(X))) |
| app(mark(X1), X2) | → | app(X1, X2) | | app(X1, mark(X2)) | → | app(X1, X2) |
| app(active(X1), X2) | → | app(X1, X2) | | app(X1, active(X2)) | → | app(X1, X2) |
| cons(mark(X1), X2) | → | cons(X1, X2) | | cons(X1, mark(X2)) | → | cons(X1, X2) |
| cons(active(X1), X2) | → | cons(X1, X2) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| from(mark(X)) | → | from(X) | | from(active(X)) | → | from(X) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| zWadr(mark(X1), X2) | → | zWadr(X1, X2) | | zWadr(X1, mark(X2)) | → | zWadr(X1, X2) |
| zWadr(active(X1), X2) | → | zWadr(X1, X2) | | zWadr(X1, active(X2)) | → | zWadr(X1, X2) |
| prefix(mark(X)) | → | prefix(X) | | prefix(active(X)) | → | prefix(X) |
Original Signature
Termination of terms over the following signature is verified: app, zWadr, s, prefix, active, mark, from, nil, cons
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| zWadr#(active(X1), X2) | → | zWadr#(X1, X2) | | zWadr#(mark(X1), X2) | → | zWadr#(X1, X2) |
Problem 11: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| zWadr#(X1, active(X2)) | → | zWadr#(X1, X2) | | zWadr#(X1, mark(X2)) | → | zWadr#(X1, X2) |
Rewrite Rules
| active(app(nil, YS)) | → | mark(YS) | | active(app(cons(X, XS), YS)) | → | mark(cons(X, app(XS, YS))) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(zWadr(nil, YS)) | → | mark(nil) |
| active(zWadr(XS, nil)) | → | mark(nil) | | active(zWadr(cons(X, XS), cons(Y, YS))) | → | mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS))) |
| active(prefix(L)) | → | mark(cons(nil, zWadr(L, prefix(L)))) | | mark(app(X1, X2)) | → | active(app(mark(X1), mark(X2))) |
| mark(nil) | → | active(nil) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(from(X)) | → | active(from(mark(X))) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(zWadr(X1, X2)) | → | active(zWadr(mark(X1), mark(X2))) | | mark(prefix(X)) | → | active(prefix(mark(X))) |
| app(mark(X1), X2) | → | app(X1, X2) | | app(X1, mark(X2)) | → | app(X1, X2) |
| app(active(X1), X2) | → | app(X1, X2) | | app(X1, active(X2)) | → | app(X1, X2) |
| cons(mark(X1), X2) | → | cons(X1, X2) | | cons(X1, mark(X2)) | → | cons(X1, X2) |
| cons(active(X1), X2) | → | cons(X1, X2) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| from(mark(X)) | → | from(X) | | from(active(X)) | → | from(X) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| zWadr(mark(X1), X2) | → | zWadr(X1, X2) | | zWadr(X1, mark(X2)) | → | zWadr(X1, X2) |
| zWadr(active(X1), X2) | → | zWadr(X1, X2) | | zWadr(X1, active(X2)) | → | zWadr(X1, X2) |
| prefix(mark(X)) | → | prefix(X) | | prefix(active(X)) | → | prefix(X) |
Original Signature
Termination of terms over the following signature is verified: app, zWadr, s, prefix, active, mark, from, cons, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| zWadr#(X1, active(X2)) | → | zWadr#(X1, X2) | | zWadr#(X1, mark(X2)) | → | zWadr#(X1, X2) |