YES
The TRS could be proven terminating. The proof took 7165 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (169ms).
| – Problem 2 was processed with processor PolynomialLinearRange4iUR (2457ms).
| | – Problem 3 was processed with processor PolynomialLinearRange4iUR (2685ms).
| | | – Problem 4 was processed with processor DependencyGraph (11ms).
| | | | – Problem 5 was processed with processor PolynomialLinearRange4iUR (1279ms).
| | | | | – Problem 6 was processed with processor DependencyGraph (4ms).
| | | | | | – Problem 7 was processed with processor PolynomialLinearRange4iUR (313ms).
| | | | | | | – Problem 8 was processed with processor PolynomialLinearRange4iUR (12ms).
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| a__from#(X) | → | mark#(X) | | a__zWadr#(cons(X, XS), cons(Y, YS)) | → | mark#(Y) |
| mark#(zWadr(X1, X2)) | → | a__zWadr#(mark(X1), mark(X2)) | | mark#(from(X)) | → | a__from#(mark(X)) |
| mark#(zWadr(X1, X2)) | → | mark#(X2) | | mark#(prefix(X)) | → | a__prefix#(mark(X)) |
| mark#(from(X)) | → | mark#(X) | | mark#(cons(X1, X2)) | → | mark#(X1) |
| mark#(prefix(X)) | → | mark#(X) | | a__zWadr#(cons(X, XS), cons(Y, YS)) | → | mark#(X) |
| mark#(app(X1, X2)) | → | a__app#(mark(X1), mark(X2)) | | a__app#(cons(X, XS), YS) | → | mark#(X) |
| mark#(app(X1, X2)) | → | mark#(X1) | | a__zWadr#(cons(X, XS), cons(Y, YS)) | → | a__app#(mark(Y), cons(mark(X), nil)) |
| mark#(zWadr(X1, X2)) | → | mark#(X1) | | mark#(s(X)) | → | mark#(X) |
| a__app#(nil, YS) | → | mark#(YS) | | mark#(app(X1, X2)) | → | mark#(X2) |
Rewrite Rules
| a__app(nil, YS) | → | mark(YS) | | a__app(cons(X, XS), YS) | → | cons(mark(X), app(XS, YS)) |
| a__from(X) | → | cons(mark(X), from(s(X))) | | a__zWadr(nil, YS) | → | nil |
| a__zWadr(XS, nil) | → | nil | | a__zWadr(cons(X, XS), cons(Y, YS)) | → | cons(a__app(mark(Y), cons(mark(X), nil)), zWadr(XS, YS)) |
| a__prefix(L) | → | cons(nil, zWadr(L, prefix(L))) | | mark(app(X1, X2)) | → | a__app(mark(X1), mark(X2)) |
| mark(from(X)) | → | a__from(mark(X)) | | mark(zWadr(X1, X2)) | → | a__zWadr(mark(X1), mark(X2)) |
| mark(prefix(X)) | → | a__prefix(mark(X)) | | mark(nil) | → | nil |
| mark(cons(X1, X2)) | → | cons(mark(X1), X2) | | mark(s(X)) | → | s(mark(X)) |
| a__app(X1, X2) | → | app(X1, X2) | | a__from(X) | → | from(X) |
| a__zWadr(X1, X2) | → | zWadr(X1, X2) | | a__prefix(X) | → | prefix(X) |
Original Signature
Termination of terms over the following signature is verified: app, zWadr, a__app, s, prefix, mark, from, a__zWadr, a__prefix, a__from, nil, cons
Strategy
The following SCCs where found
| a__from#(X) → mark#(X) | a__zWadr#(cons(X, XS), cons(Y, YS)) → mark#(Y) |
| mark#(zWadr(X1, X2)) → a__zWadr#(mark(X1), mark(X2)) | mark#(from(X)) → a__from#(mark(X)) |
| mark#(zWadr(X1, X2)) → mark#(X2) | mark#(from(X)) → mark#(X) |
| mark#(cons(X1, X2)) → mark#(X1) | mark#(prefix(X)) → mark#(X) |
| a__zWadr#(cons(X, XS), cons(Y, YS)) → mark#(X) | mark#(app(X1, X2)) → a__app#(mark(X1), mark(X2)) |
| a__app#(cons(X, XS), YS) → mark#(X) | mark#(app(X1, X2)) → mark#(X1) |
| a__zWadr#(cons(X, XS), cons(Y, YS)) → a__app#(mark(Y), cons(mark(X), nil)) | mark#(zWadr(X1, X2)) → mark#(X1) |
| mark#(s(X)) → mark#(X) | a__app#(nil, YS) → mark#(YS) |
| mark#(app(X1, X2)) → mark#(X2) |
Problem 2: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| a__from#(X) | → | mark#(X) | | mark#(zWadr(X1, X2)) | → | a__zWadr#(mark(X1), mark(X2)) |
| a__zWadr#(cons(X, XS), cons(Y, YS)) | → | mark#(Y) | | mark#(from(X)) | → | a__from#(mark(X)) |
| mark#(zWadr(X1, X2)) | → | mark#(X2) | | mark#(from(X)) | → | mark#(X) |
| mark#(cons(X1, X2)) | → | mark#(X1) | | mark#(prefix(X)) | → | mark#(X) |
| mark#(app(X1, X2)) | → | a__app#(mark(X1), mark(X2)) | | a__zWadr#(cons(X, XS), cons(Y, YS)) | → | mark#(X) |
| mark#(app(X1, X2)) | → | mark#(X1) | | a__app#(cons(X, XS), YS) | → | mark#(X) |
| a__zWadr#(cons(X, XS), cons(Y, YS)) | → | a__app#(mark(Y), cons(mark(X), nil)) | | mark#(zWadr(X1, X2)) | → | mark#(X1) |
| mark#(s(X)) | → | mark#(X) | | a__app#(nil, YS) | → | mark#(YS) |
| mark#(app(X1, X2)) | → | mark#(X2) |
Rewrite Rules
| a__app(nil, YS) | → | mark(YS) | | a__app(cons(X, XS), YS) | → | cons(mark(X), app(XS, YS)) |
| a__from(X) | → | cons(mark(X), from(s(X))) | | a__zWadr(nil, YS) | → | nil |
| a__zWadr(XS, nil) | → | nil | | a__zWadr(cons(X, XS), cons(Y, YS)) | → | cons(a__app(mark(Y), cons(mark(X), nil)), zWadr(XS, YS)) |
| a__prefix(L) | → | cons(nil, zWadr(L, prefix(L))) | | mark(app(X1, X2)) | → | a__app(mark(X1), mark(X2)) |
| mark(from(X)) | → | a__from(mark(X)) | | mark(zWadr(X1, X2)) | → | a__zWadr(mark(X1), mark(X2)) |
| mark(prefix(X)) | → | a__prefix(mark(X)) | | mark(nil) | → | nil |
| mark(cons(X1, X2)) | → | cons(mark(X1), X2) | | mark(s(X)) | → | s(mark(X)) |
| a__app(X1, X2) | → | app(X1, X2) | | a__from(X) | → | from(X) |
| a__zWadr(X1, X2) | → | zWadr(X1, X2) | | a__prefix(X) | → | prefix(X) |
Original Signature
Termination of terms over the following signature is verified: app, zWadr, a__app, s, prefix, mark, from, a__zWadr, a__prefix, a__from, nil, cons
Strategy
Polynomial Interpretation
- a__app(x,y): y + x
- a__app#(x,y): 2y + 2x
- a__from(x): x
- a__from#(x): 2x
- a__prefix(x): x + 1
- a__zWadr(x,y): y + x
- a__zWadr#(x,y): 2y + 2x
- app(x,y): y + x
- cons(x,y): x
- from(x): x
- mark(x): x
- mark#(x): 2x
- nil: 1
- prefix(x): x + 1
- s(x): x
- zWadr(x,y): y + x
Improved Usable rules
| a__prefix(L) | → | cons(nil, zWadr(L, prefix(L))) | | a__zWadr(nil, YS) | → | nil |
| mark(cons(X1, X2)) | → | cons(mark(X1), X2) | | a__from(X) | → | from(X) |
| a__from(X) | → | cons(mark(X), from(s(X))) | | a__app(X1, X2) | → | app(X1, X2) |
| mark(app(X1, X2)) | → | a__app(mark(X1), mark(X2)) | | a__zWadr(XS, nil) | → | nil |
| a__zWadr(cons(X, XS), cons(Y, YS)) | → | cons(a__app(mark(Y), cons(mark(X), nil)), zWadr(XS, YS)) | | a__zWadr(X1, X2) | → | zWadr(X1, X2) |
| mark(s(X)) | → | s(mark(X)) | | mark(from(X)) | → | a__from(mark(X)) |
| mark(zWadr(X1, X2)) | → | a__zWadr(mark(X1), mark(X2)) | | mark(nil) | → | nil |
| a__app(nil, YS) | → | mark(YS) | | mark(prefix(X)) | → | a__prefix(mark(X)) |
| a__app(cons(X, XS), YS) | → | cons(mark(X), app(XS, YS)) | | a__prefix(X) | → | prefix(X) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| mark#(prefix(X)) | → | mark#(X) | | a__app#(nil, YS) | → | mark#(YS) |
Problem 3: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| a__from#(X) | → | mark#(X) | | mark#(zWadr(X1, X2)) | → | a__zWadr#(mark(X1), mark(X2)) |
| a__zWadr#(cons(X, XS), cons(Y, YS)) | → | mark#(Y) | | mark#(from(X)) | → | a__from#(mark(X)) |
| mark#(zWadr(X1, X2)) | → | mark#(X2) | | mark#(from(X)) | → | mark#(X) |
| mark#(cons(X1, X2)) | → | mark#(X1) | | mark#(app(X1, X2)) | → | a__app#(mark(X1), mark(X2)) |
| a__zWadr#(cons(X, XS), cons(Y, YS)) | → | mark#(X) | | mark#(app(X1, X2)) | → | mark#(X1) |
| a__app#(cons(X, XS), YS) | → | mark#(X) | | a__zWadr#(cons(X, XS), cons(Y, YS)) | → | a__app#(mark(Y), cons(mark(X), nil)) |
| mark#(zWadr(X1, X2)) | → | mark#(X1) | | mark#(s(X)) | → | mark#(X) |
| mark#(app(X1, X2)) | → | mark#(X2) |
Rewrite Rules
| a__app(nil, YS) | → | mark(YS) | | a__app(cons(X, XS), YS) | → | cons(mark(X), app(XS, YS)) |
| a__from(X) | → | cons(mark(X), from(s(X))) | | a__zWadr(nil, YS) | → | nil |
| a__zWadr(XS, nil) | → | nil | | a__zWadr(cons(X, XS), cons(Y, YS)) | → | cons(a__app(mark(Y), cons(mark(X), nil)), zWadr(XS, YS)) |
| a__prefix(L) | → | cons(nil, zWadr(L, prefix(L))) | | mark(app(X1, X2)) | → | a__app(mark(X1), mark(X2)) |
| mark(from(X)) | → | a__from(mark(X)) | | mark(zWadr(X1, X2)) | → | a__zWadr(mark(X1), mark(X2)) |
| mark(prefix(X)) | → | a__prefix(mark(X)) | | mark(nil) | → | nil |
| mark(cons(X1, X2)) | → | cons(mark(X1), X2) | | mark(s(X)) | → | s(mark(X)) |
| a__app(X1, X2) | → | app(X1, X2) | | a__from(X) | → | from(X) |
| a__zWadr(X1, X2) | → | zWadr(X1, X2) | | a__prefix(X) | → | prefix(X) |
Original Signature
Termination of terms over the following signature is verified: app, zWadr, a__app, s, prefix, mark, a__zWadr, from, a__prefix, a__from, cons, nil
Strategy
Polynomial Interpretation
- a__app(x,y): y + x
- a__app#(x,y): x
- a__from(x): x + 2
- a__from#(x): x + 2
- a__prefix(x): 0
- a__zWadr(x,y): y + 2x
- a__zWadr#(x,y): y + x
- app(x,y): y + x
- cons(x,y): x
- from(x): x + 2
- mark(x): x
- mark#(x): x
- nil: 0
- prefix(x): 0
- s(x): x
- zWadr(x,y): y + 2x
Improved Usable rules
| a__prefix(L) | → | cons(nil, zWadr(L, prefix(L))) | | a__zWadr(nil, YS) | → | nil |
| mark(cons(X1, X2)) | → | cons(mark(X1), X2) | | a__from(X) | → | from(X) |
| a__from(X) | → | cons(mark(X), from(s(X))) | | a__app(X1, X2) | → | app(X1, X2) |
| mark(app(X1, X2)) | → | a__app(mark(X1), mark(X2)) | | a__zWadr(XS, nil) | → | nil |
| a__zWadr(cons(X, XS), cons(Y, YS)) | → | cons(a__app(mark(Y), cons(mark(X), nil)), zWadr(XS, YS)) | | a__zWadr(X1, X2) | → | zWadr(X1, X2) |
| mark(s(X)) | → | s(mark(X)) | | mark(from(X)) | → | a__from(mark(X)) |
| mark(zWadr(X1, X2)) | → | a__zWadr(mark(X1), mark(X2)) | | mark(nil) | → | nil |
| a__app(nil, YS) | → | mark(YS) | | mark(prefix(X)) | → | a__prefix(mark(X)) |
| a__app(cons(X, XS), YS) | → | cons(mark(X), app(XS, YS)) | | a__prefix(X) | → | prefix(X) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| a__from#(X) | → | mark#(X) | | mark#(from(X)) | → | mark#(X) |
Problem 4: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| mark#(zWadr(X1, X2)) | → | a__zWadr#(mark(X1), mark(X2)) | | a__zWadr#(cons(X, XS), cons(Y, YS)) | → | mark#(Y) |
| mark#(from(X)) | → | a__from#(mark(X)) | | mark#(zWadr(X1, X2)) | → | mark#(X2) |
| mark#(cons(X1, X2)) | → | mark#(X1) | | mark#(app(X1, X2)) | → | a__app#(mark(X1), mark(X2)) |
| a__zWadr#(cons(X, XS), cons(Y, YS)) | → | mark#(X) | | mark#(app(X1, X2)) | → | mark#(X1) |
| a__app#(cons(X, XS), YS) | → | mark#(X) | | a__zWadr#(cons(X, XS), cons(Y, YS)) | → | a__app#(mark(Y), cons(mark(X), nil)) |
| mark#(zWadr(X1, X2)) | → | mark#(X1) | | mark#(s(X)) | → | mark#(X) |
| mark#(app(X1, X2)) | → | mark#(X2) |
Rewrite Rules
| a__app(nil, YS) | → | mark(YS) | | a__app(cons(X, XS), YS) | → | cons(mark(X), app(XS, YS)) |
| a__from(X) | → | cons(mark(X), from(s(X))) | | a__zWadr(nil, YS) | → | nil |
| a__zWadr(XS, nil) | → | nil | | a__zWadr(cons(X, XS), cons(Y, YS)) | → | cons(a__app(mark(Y), cons(mark(X), nil)), zWadr(XS, YS)) |
| a__prefix(L) | → | cons(nil, zWadr(L, prefix(L))) | | mark(app(X1, X2)) | → | a__app(mark(X1), mark(X2)) |
| mark(from(X)) | → | a__from(mark(X)) | | mark(zWadr(X1, X2)) | → | a__zWadr(mark(X1), mark(X2)) |
| mark(prefix(X)) | → | a__prefix(mark(X)) | | mark(nil) | → | nil |
| mark(cons(X1, X2)) | → | cons(mark(X1), X2) | | mark(s(X)) | → | s(mark(X)) |
| a__app(X1, X2) | → | app(X1, X2) | | a__from(X) | → | from(X) |
| a__zWadr(X1, X2) | → | zWadr(X1, X2) | | a__prefix(X) | → | prefix(X) |
Original Signature
Termination of terms over the following signature is verified: app, zWadr, a__app, s, prefix, mark, from, a__zWadr, a__prefix, a__from, nil, cons
Strategy
The following SCCs where found
| a__zWadr#(cons(X, XS), cons(Y, YS)) → mark#(Y) | mark#(zWadr(X1, X2)) → a__zWadr#(mark(X1), mark(X2)) |
| mark#(app(X1, X2)) → a__app#(mark(X1), mark(X2)) | a__zWadr#(cons(X, XS), cons(Y, YS)) → mark#(X) |
| a__app#(cons(X, XS), YS) → mark#(X) | mark#(zWadr(X1, X2)) → mark#(X2) |
| mark#(app(X1, X2)) → mark#(X1) | a__zWadr#(cons(X, XS), cons(Y, YS)) → a__app#(mark(Y), cons(mark(X), nil)) |
| mark#(cons(X1, X2)) → mark#(X1) | mark#(zWadr(X1, X2)) → mark#(X1) |
| mark#(s(X)) → mark#(X) | mark#(app(X1, X2)) → mark#(X2) |
Problem 5: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| a__zWadr#(cons(X, XS), cons(Y, YS)) | → | mark#(Y) | | mark#(zWadr(X1, X2)) | → | a__zWadr#(mark(X1), mark(X2)) |
| mark#(app(X1, X2)) | → | a__app#(mark(X1), mark(X2)) | | a__zWadr#(cons(X, XS), cons(Y, YS)) | → | mark#(X) |
| a__app#(cons(X, XS), YS) | → | mark#(X) | | mark#(app(X1, X2)) | → | mark#(X1) |
| mark#(zWadr(X1, X2)) | → | mark#(X2) | | a__zWadr#(cons(X, XS), cons(Y, YS)) | → | a__app#(mark(Y), cons(mark(X), nil)) |
| mark#(cons(X1, X2)) | → | mark#(X1) | | mark#(zWadr(X1, X2)) | → | mark#(X1) |
| mark#(s(X)) | → | mark#(X) | | mark#(app(X1, X2)) | → | mark#(X2) |
Rewrite Rules
| a__app(nil, YS) | → | mark(YS) | | a__app(cons(X, XS), YS) | → | cons(mark(X), app(XS, YS)) |
| a__from(X) | → | cons(mark(X), from(s(X))) | | a__zWadr(nil, YS) | → | nil |
| a__zWadr(XS, nil) | → | nil | | a__zWadr(cons(X, XS), cons(Y, YS)) | → | cons(a__app(mark(Y), cons(mark(X), nil)), zWadr(XS, YS)) |
| a__prefix(L) | → | cons(nil, zWadr(L, prefix(L))) | | mark(app(X1, X2)) | → | a__app(mark(X1), mark(X2)) |
| mark(from(X)) | → | a__from(mark(X)) | | mark(zWadr(X1, X2)) | → | a__zWadr(mark(X1), mark(X2)) |
| mark(prefix(X)) | → | a__prefix(mark(X)) | | mark(nil) | → | nil |
| mark(cons(X1, X2)) | → | cons(mark(X1), X2) | | mark(s(X)) | → | s(mark(X)) |
| a__app(X1, X2) | → | app(X1, X2) | | a__from(X) | → | from(X) |
| a__zWadr(X1, X2) | → | zWadr(X1, X2) | | a__prefix(X) | → | prefix(X) |
Original Signature
Termination of terms over the following signature is verified: app, zWadr, a__app, s, prefix, mark, from, a__zWadr, a__prefix, a__from, nil, cons
Strategy
Polynomial Interpretation
- a__app(x,y): y + x
- a__app#(x,y): x
- a__from(x): x
- a__prefix(x): 0
- a__zWadr(x,y): y + x + 2
- a__zWadr#(x,y): y + x
- app(x,y): y + x
- cons(x,y): x
- from(x): x
- mark(x): x
- mark#(x): x
- nil: 0
- prefix(x): 0
- s(x): 2x + 2
- zWadr(x,y): y + x + 2
Improved Usable rules
| a__prefix(L) | → | cons(nil, zWadr(L, prefix(L))) | | a__zWadr(nil, YS) | → | nil |
| mark(cons(X1, X2)) | → | cons(mark(X1), X2) | | a__from(X) | → | from(X) |
| a__from(X) | → | cons(mark(X), from(s(X))) | | a__app(X1, X2) | → | app(X1, X2) |
| mark(app(X1, X2)) | → | a__app(mark(X1), mark(X2)) | | a__zWadr(XS, nil) | → | nil |
| a__zWadr(cons(X, XS), cons(Y, YS)) | → | cons(a__app(mark(Y), cons(mark(X), nil)), zWadr(XS, YS)) | | a__zWadr(X1, X2) | → | zWadr(X1, X2) |
| mark(s(X)) | → | s(mark(X)) | | mark(from(X)) | → | a__from(mark(X)) |
| mark(zWadr(X1, X2)) | → | a__zWadr(mark(X1), mark(X2)) | | mark(nil) | → | nil |
| a__app(nil, YS) | → | mark(YS) | | mark(prefix(X)) | → | a__prefix(mark(X)) |
| a__app(cons(X, XS), YS) | → | cons(mark(X), app(XS, YS)) | | a__prefix(X) | → | prefix(X) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| mark#(zWadr(X1, X2)) | → | a__zWadr#(mark(X1), mark(X2)) | | mark#(zWadr(X1, X2)) | → | mark#(X2) |
| mark#(zWadr(X1, X2)) | → | mark#(X1) | | mark#(s(X)) | → | mark#(X) |
Problem 6: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| a__zWadr#(cons(X, XS), cons(Y, YS)) | → | mark#(Y) | | a__zWadr#(cons(X, XS), cons(Y, YS)) | → | mark#(X) |
| mark#(app(X1, X2)) | → | a__app#(mark(X1), mark(X2)) | | mark#(app(X1, X2)) | → | mark#(X1) |
| a__app#(cons(X, XS), YS) | → | mark#(X) | | a__zWadr#(cons(X, XS), cons(Y, YS)) | → | a__app#(mark(Y), cons(mark(X), nil)) |
| mark#(cons(X1, X2)) | → | mark#(X1) | | mark#(app(X1, X2)) | → | mark#(X2) |
Rewrite Rules
| a__app(nil, YS) | → | mark(YS) | | a__app(cons(X, XS), YS) | → | cons(mark(X), app(XS, YS)) |
| a__from(X) | → | cons(mark(X), from(s(X))) | | a__zWadr(nil, YS) | → | nil |
| a__zWadr(XS, nil) | → | nil | | a__zWadr(cons(X, XS), cons(Y, YS)) | → | cons(a__app(mark(Y), cons(mark(X), nil)), zWadr(XS, YS)) |
| a__prefix(L) | → | cons(nil, zWadr(L, prefix(L))) | | mark(app(X1, X2)) | → | a__app(mark(X1), mark(X2)) |
| mark(from(X)) | → | a__from(mark(X)) | | mark(zWadr(X1, X2)) | → | a__zWadr(mark(X1), mark(X2)) |
| mark(prefix(X)) | → | a__prefix(mark(X)) | | mark(nil) | → | nil |
| mark(cons(X1, X2)) | → | cons(mark(X1), X2) | | mark(s(X)) | → | s(mark(X)) |
| a__app(X1, X2) | → | app(X1, X2) | | a__from(X) | → | from(X) |
| a__zWadr(X1, X2) | → | zWadr(X1, X2) | | a__prefix(X) | → | prefix(X) |
Original Signature
Termination of terms over the following signature is verified: app, zWadr, a__app, s, prefix, mark, a__zWadr, from, a__prefix, a__from, cons, nil
Strategy
The following SCCs where found
| mark#(app(X1, X2)) → a__app#(mark(X1), mark(X2)) | a__app#(cons(X, XS), YS) → mark#(X) |
| mark#(app(X1, X2)) → mark#(X1) | mark#(cons(X1, X2)) → mark#(X1) |
| mark#(app(X1, X2)) → mark#(X2) |
Problem 7: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| mark#(app(X1, X2)) | → | a__app#(mark(X1), mark(X2)) | | a__app#(cons(X, XS), YS) | → | mark#(X) |
| mark#(app(X1, X2)) | → | mark#(X1) | | mark#(cons(X1, X2)) | → | mark#(X1) |
| mark#(app(X1, X2)) | → | mark#(X2) |
Rewrite Rules
| a__app(nil, YS) | → | mark(YS) | | a__app(cons(X, XS), YS) | → | cons(mark(X), app(XS, YS)) |
| a__from(X) | → | cons(mark(X), from(s(X))) | | a__zWadr(nil, YS) | → | nil |
| a__zWadr(XS, nil) | → | nil | | a__zWadr(cons(X, XS), cons(Y, YS)) | → | cons(a__app(mark(Y), cons(mark(X), nil)), zWadr(XS, YS)) |
| a__prefix(L) | → | cons(nil, zWadr(L, prefix(L))) | | mark(app(X1, X2)) | → | a__app(mark(X1), mark(X2)) |
| mark(from(X)) | → | a__from(mark(X)) | | mark(zWadr(X1, X2)) | → | a__zWadr(mark(X1), mark(X2)) |
| mark(prefix(X)) | → | a__prefix(mark(X)) | | mark(nil) | → | nil |
| mark(cons(X1, X2)) | → | cons(mark(X1), X2) | | mark(s(X)) | → | s(mark(X)) |
| a__app(X1, X2) | → | app(X1, X2) | | a__from(X) | → | from(X) |
| a__zWadr(X1, X2) | → | zWadr(X1, X2) | | a__prefix(X) | → | prefix(X) |
Original Signature
Termination of terms over the following signature is verified: app, zWadr, a__app, s, prefix, mark, a__zWadr, from, a__prefix, a__from, cons, nil
Strategy
Polynomial Interpretation
- a__app(x,y): y + x + 1
- a__app#(x,y): 2x + 1
- a__from(x): 2x
- a__prefix(x): x
- a__zWadr(x,y): 2y + 2x + 2
- app(x,y): y + x + 1
- cons(x,y): 2x
- from(x): 2x
- mark(x): x
- mark#(x): 2x
- nil: 0
- prefix(x): x
- s(x): 0
- zWadr(x,y): 2y + 2x + 2
Improved Usable rules
| a__prefix(L) | → | cons(nil, zWadr(L, prefix(L))) | | a__zWadr(nil, YS) | → | nil |
| mark(cons(X1, X2)) | → | cons(mark(X1), X2) | | a__from(X) | → | from(X) |
| a__from(X) | → | cons(mark(X), from(s(X))) | | a__app(X1, X2) | → | app(X1, X2) |
| mark(app(X1, X2)) | → | a__app(mark(X1), mark(X2)) | | a__zWadr(XS, nil) | → | nil |
| a__zWadr(cons(X, XS), cons(Y, YS)) | → | cons(a__app(mark(Y), cons(mark(X), nil)), zWadr(XS, YS)) | | a__zWadr(X1, X2) | → | zWadr(X1, X2) |
| mark(s(X)) | → | s(mark(X)) | | mark(from(X)) | → | a__from(mark(X)) |
| mark(zWadr(X1, X2)) | → | a__zWadr(mark(X1), mark(X2)) | | mark(nil) | → | nil |
| a__app(nil, YS) | → | mark(YS) | | mark(prefix(X)) | → | a__prefix(mark(X)) |
| a__app(cons(X, XS), YS) | → | cons(mark(X), app(XS, YS)) | | a__prefix(X) | → | prefix(X) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| mark#(app(X1, X2)) | → | a__app#(mark(X1), mark(X2)) | | mark#(app(X1, X2)) | → | mark#(X1) |
| a__app#(cons(X, XS), YS) | → | mark#(X) | | mark#(app(X1, X2)) | → | mark#(X2) |
Problem 8: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| mark#(cons(X1, X2)) | → | mark#(X1) |
Rewrite Rules
| a__app(nil, YS) | → | mark(YS) | | a__app(cons(X, XS), YS) | → | cons(mark(X), app(XS, YS)) |
| a__from(X) | → | cons(mark(X), from(s(X))) | | a__zWadr(nil, YS) | → | nil |
| a__zWadr(XS, nil) | → | nil | | a__zWadr(cons(X, XS), cons(Y, YS)) | → | cons(a__app(mark(Y), cons(mark(X), nil)), zWadr(XS, YS)) |
| a__prefix(L) | → | cons(nil, zWadr(L, prefix(L))) | | mark(app(X1, X2)) | → | a__app(mark(X1), mark(X2)) |
| mark(from(X)) | → | a__from(mark(X)) | | mark(zWadr(X1, X2)) | → | a__zWadr(mark(X1), mark(X2)) |
| mark(prefix(X)) | → | a__prefix(mark(X)) | | mark(nil) | → | nil |
| mark(cons(X1, X2)) | → | cons(mark(X1), X2) | | mark(s(X)) | → | s(mark(X)) |
| a__app(X1, X2) | → | app(X1, X2) | | a__from(X) | → | from(X) |
| a__zWadr(X1, X2) | → | zWadr(X1, X2) | | a__prefix(X) | → | prefix(X) |
Original Signature
Termination of terms over the following signature is verified: app, zWadr, a__app, s, prefix, mark, from, a__zWadr, a__prefix, a__from, nil, cons
Strategy
Polynomial Interpretation
- a__app(x,y): 0
- a__from(x): 0
- a__prefix(x): 0
- a__zWadr(x,y): 0
- app(x,y): 0
- cons(x,y): x + 2
- from(x): 0
- mark(x): 0
- mark#(x): x + 1
- nil: 0
- prefix(x): 0
- s(x): 0
- 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) |