YES
The TRS could be proven terminating. The proof took 29082 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (303ms).
| – Problem 2 was processed with processor SubtermCriterion (1ms).
| – Problem 3 was processed with processor PolynomialLinearRange4iUR (6122ms).
| | – Problem 8 was processed with processor PolynomialLinearRange4iUR (4276ms).
| | | – Problem 9 was processed with processor PolynomialLinearRange4iUR (4579ms).
| | | | – Problem 10 was processed with processor PolynomialLinearRange4iUR (3200ms).
| | | | | – Problem 11 was processed with processor PolynomialLinearRange4iUR (2957ms).
| | | | | | – Problem 12 was processed with processor PolynomialLinearRange4iUR (2792ms).
| | | | | | | – Problem 13 was processed with processor PolynomialLinearRange4iUR (2283ms).
| | | | | | | | – Problem 14 was processed with processor PolynomialLinearRange4iUR (2300ms).
| | | | | | | | | – Problem 15 was processed with processor DependencyGraph (0ms).
| – Problem 4 was processed with processor SubtermCriterion (0ms).
| – Problem 5 was processed with processor SubtermCriterion (10ms).
| – Problem 6 was processed with processor SubtermCriterion (1ms).
| | – Problem 7 was processed with processor SubtermCriterion (1ms).
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| mark#(__(X1, X2)) | → | __#(mark(X1), mark(X2)) | | mark#(U11(X)) | → | U11#(mark(X)) |
| mark#(isNePal(X)) | → | isNePal#(mark(X)) | | U11#(active(X)) | → | U11#(X) |
| active#(__(__(X, Y), Z)) | → | mark#(__(X, __(Y, Z))) | | mark#(tt) | → | active#(tt) |
| active#(__(__(X, Y), Z)) | → | __#(X, __(Y, Z)) | | active#(__(X, nil)) | → | mark#(X) |
| mark#(U11(X)) | → | mark#(X) | | U12#(mark(X)) | → | U12#(X) |
| __#(X1, active(X2)) | → | __#(X1, X2) | | __#(active(X1), X2) | → | __#(X1, X2) |
| active#(U11(tt)) | → | mark#(U12(tt)) | | mark#(nil) | → | active#(nil) |
| active#(isNePal(__(I, __(P, I)))) | → | U11#(tt) | | mark#(U11(X)) | → | active#(U11(mark(X))) |
| active#(isNePal(__(I, __(P, I)))) | → | mark#(U11(tt)) | | mark#(isNePal(X)) | → | mark#(X) |
| mark#(__(X1, X2)) | → | mark#(X2) | | U11#(mark(X)) | → | U11#(X) |
| __#(mark(X1), X2) | → | __#(X1, X2) | | mark#(U12(X)) | → | mark#(X) |
| isNePal#(active(X)) | → | isNePal#(X) | | mark#(U12(X)) | → | U12#(mark(X)) |
| active#(__(nil, X)) | → | mark#(X) | | active#(U11(tt)) | → | U12#(tt) |
| active#(__(__(X, Y), Z)) | → | __#(Y, Z) | | __#(X1, mark(X2)) | → | __#(X1, X2) |
| mark#(__(X1, X2)) | → | active#(__(mark(X1), mark(X2))) | | isNePal#(mark(X)) | → | isNePal#(X) |
| mark#(isNePal(X)) | → | active#(isNePal(mark(X))) | | mark#(U12(X)) | → | active#(U12(mark(X))) |
| active#(U12(tt)) | → | mark#(tt) | | U12#(active(X)) | → | U12#(X) |
| mark#(__(X1, X2)) | → | mark#(X1) |
Rewrite Rules
| active(__(__(X, Y), Z)) | → | mark(__(X, __(Y, Z))) | | active(__(X, nil)) | → | mark(X) |
| active(__(nil, X)) | → | mark(X) | | active(U11(tt)) | → | mark(U12(tt)) |
| active(U12(tt)) | → | mark(tt) | | active(isNePal(__(I, __(P, I)))) | → | mark(U11(tt)) |
| mark(__(X1, X2)) | → | active(__(mark(X1), mark(X2))) | | mark(nil) | → | active(nil) |
| mark(U11(X)) | → | active(U11(mark(X))) | | mark(tt) | → | active(tt) |
| mark(U12(X)) | → | active(U12(mark(X))) | | mark(isNePal(X)) | → | active(isNePal(mark(X))) |
| __(mark(X1), X2) | → | __(X1, X2) | | __(X1, mark(X2)) | → | __(X1, X2) |
| __(active(X1), X2) | → | __(X1, X2) | | __(X1, active(X2)) | → | __(X1, X2) |
| U11(mark(X)) | → | U11(X) | | U11(active(X)) | → | U11(X) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNePal(mark(X)) | → | isNePal(X) | | isNePal(active(X)) | → | isNePal(X) |
Original Signature
Termination of terms over the following signature is verified: tt, isNePal, active, __, U11, mark, U12, nil
Strategy
The following SCCs where found
| mark#(U12(X)) → mark#(X) | active#(__(__(X, Y), Z)) → mark#(__(X, __(Y, Z))) |
| active#(__(X, nil)) → mark#(X) | active#(__(nil, X)) → mark#(X) |
| mark#(U11(X)) → mark#(X) | mark#(__(X1, X2)) → active#(__(mark(X1), mark(X2))) |
| mark#(isNePal(X)) → active#(isNePal(mark(X))) | mark#(U12(X)) → active#(U12(mark(X))) |
| active#(U11(tt)) → mark#(U12(tt)) | mark#(U11(X)) → active#(U11(mark(X))) |
| mark#(__(X1, X2)) → mark#(X1) | active#(isNePal(__(I, __(P, I)))) → mark#(U11(tt)) |
| mark#(isNePal(X)) → mark#(X) | mark#(__(X1, X2)) → mark#(X2) |
| U12#(active(X)) → U12#(X) | U12#(mark(X)) → U12#(X) |
| __#(active(X1), X2) → __#(X1, X2) | __#(mark(X1), X2) → __#(X1, X2) |
| __#(X1, mark(X2)) → __#(X1, X2) | __#(X1, active(X2)) → __#(X1, X2) |
| isNePal#(active(X)) → isNePal#(X) | isNePal#(mark(X)) → isNePal#(X) |
| U11#(active(X)) → U11#(X) | U11#(mark(X)) → U11#(X) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U12#(active(X)) | → | U12#(X) | | U12#(mark(X)) | → | U12#(X) |
Rewrite Rules
| active(__(__(X, Y), Z)) | → | mark(__(X, __(Y, Z))) | | active(__(X, nil)) | → | mark(X) |
| active(__(nil, X)) | → | mark(X) | | active(U11(tt)) | → | mark(U12(tt)) |
| active(U12(tt)) | → | mark(tt) | | active(isNePal(__(I, __(P, I)))) | → | mark(U11(tt)) |
| mark(__(X1, X2)) | → | active(__(mark(X1), mark(X2))) | | mark(nil) | → | active(nil) |
| mark(U11(X)) | → | active(U11(mark(X))) | | mark(tt) | → | active(tt) |
| mark(U12(X)) | → | active(U12(mark(X))) | | mark(isNePal(X)) | → | active(isNePal(mark(X))) |
| __(mark(X1), X2) | → | __(X1, X2) | | __(X1, mark(X2)) | → | __(X1, X2) |
| __(active(X1), X2) | → | __(X1, X2) | | __(X1, active(X2)) | → | __(X1, X2) |
| U11(mark(X)) | → | U11(X) | | U11(active(X)) | → | U11(X) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNePal(mark(X)) | → | isNePal(X) | | isNePal(active(X)) | → | isNePal(X) |
Original Signature
Termination of terms over the following signature is verified: tt, isNePal, active, __, U11, mark, U12, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U12#(active(X)) | → | U12#(X) | | U12#(mark(X)) | → | U12#(X) |
Problem 3: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| mark#(U12(X)) | → | mark#(X) | | active#(__(__(X, Y), Z)) | → | mark#(__(X, __(Y, Z))) |
| active#(__(X, nil)) | → | mark#(X) | | active#(__(nil, X)) | → | mark#(X) |
| mark#(U11(X)) | → | mark#(X) | | mark#(__(X1, X2)) | → | active#(__(mark(X1), mark(X2))) |
| mark#(isNePal(X)) | → | active#(isNePal(mark(X))) | | mark#(U12(X)) | → | active#(U12(mark(X))) |
| active#(U11(tt)) | → | mark#(U12(tt)) | | mark#(__(X1, X2)) | → | mark#(X1) |
| mark#(U11(X)) | → | active#(U11(mark(X))) | | active#(isNePal(__(I, __(P, I)))) | → | mark#(U11(tt)) |
| mark#(isNePal(X)) | → | mark#(X) | | mark#(__(X1, X2)) | → | mark#(X2) |
Rewrite Rules
| active(__(__(X, Y), Z)) | → | mark(__(X, __(Y, Z))) | | active(__(X, nil)) | → | mark(X) |
| active(__(nil, X)) | → | mark(X) | | active(U11(tt)) | → | mark(U12(tt)) |
| active(U12(tt)) | → | mark(tt) | | active(isNePal(__(I, __(P, I)))) | → | mark(U11(tt)) |
| mark(__(X1, X2)) | → | active(__(mark(X1), mark(X2))) | | mark(nil) | → | active(nil) |
| mark(U11(X)) | → | active(U11(mark(X))) | | mark(tt) | → | active(tt) |
| mark(U12(X)) | → | active(U12(mark(X))) | | mark(isNePal(X)) | → | active(isNePal(mark(X))) |
| __(mark(X1), X2) | → | __(X1, X2) | | __(X1, mark(X2)) | → | __(X1, X2) |
| __(active(X1), X2) | → | __(X1, X2) | | __(X1, active(X2)) | → | __(X1, X2) |
| U11(mark(X)) | → | U11(X) | | U11(active(X)) | → | U11(X) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNePal(mark(X)) | → | isNePal(X) | | isNePal(active(X)) | → | isNePal(X) |
Original Signature
Termination of terms over the following signature is verified: tt, isNePal, active, __, U11, mark, U12, nil
Strategy
Polynomial Interpretation
- U11(x): 1
- U12(x): 0
- __(x,y): 1
- active(x): 3
- active#(x): 2x
- isNePal(x): 1
- mark(x): 0
- mark#(x): 2
- nil: 1
- tt: 2
Improved Usable rules
| isNePal(active(X)) | → | isNePal(X) | | __(mark(X1), X2) | → | __(X1, X2) |
| U12(active(X)) | → | U12(X) | | U11(mark(X)) | → | U11(X) |
| isNePal(mark(X)) | → | isNePal(X) | | __(active(X1), X2) | → | __(X1, X2) |
| __(X1, mark(X2)) | → | __(X1, X2) | | __(X1, active(X2)) | → | __(X1, X2) |
| U11(active(X)) | → | U11(X) | | U12(mark(X)) | → | U12(X) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| mark#(U12(X)) | → | active#(U12(mark(X))) |
Problem 8: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| mark#(U12(X)) | → | mark#(X) | | active#(__(__(X, Y), Z)) | → | mark#(__(X, __(Y, Z))) |
| active#(__(X, nil)) | → | mark#(X) | | active#(__(nil, X)) | → | mark#(X) |
| mark#(U11(X)) | → | mark#(X) | | mark#(__(X1, X2)) | → | active#(__(mark(X1), mark(X2))) |
| mark#(isNePal(X)) | → | active#(isNePal(mark(X))) | | active#(U11(tt)) | → | mark#(U12(tt)) |
| mark#(__(X1, X2)) | → | mark#(X1) | | mark#(U11(X)) | → | active#(U11(mark(X))) |
| active#(isNePal(__(I, __(P, I)))) | → | mark#(U11(tt)) | | mark#(isNePal(X)) | → | mark#(X) |
| mark#(__(X1, X2)) | → | mark#(X2) |
Rewrite Rules
| active(__(__(X, Y), Z)) | → | mark(__(X, __(Y, Z))) | | active(__(X, nil)) | → | mark(X) |
| active(__(nil, X)) | → | mark(X) | | active(U11(tt)) | → | mark(U12(tt)) |
| active(U12(tt)) | → | mark(tt) | | active(isNePal(__(I, __(P, I)))) | → | mark(U11(tt)) |
| mark(__(X1, X2)) | → | active(__(mark(X1), mark(X2))) | | mark(nil) | → | active(nil) |
| mark(U11(X)) | → | active(U11(mark(X))) | | mark(tt) | → | active(tt) |
| mark(U12(X)) | → | active(U12(mark(X))) | | mark(isNePal(X)) | → | active(isNePal(mark(X))) |
| __(mark(X1), X2) | → | __(X1, X2) | | __(X1, mark(X2)) | → | __(X1, X2) |
| __(active(X1), X2) | → | __(X1, X2) | | __(X1, active(X2)) | → | __(X1, X2) |
| U11(mark(X)) | → | U11(X) | | U11(active(X)) | → | U11(X) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNePal(mark(X)) | → | isNePal(X) | | isNePal(active(X)) | → | isNePal(X) |
Original Signature
Termination of terms over the following signature is verified: tt, isNePal, U11, __, active, U12, mark, nil
Strategy
Polynomial Interpretation
- U11(x): 2x + 1
- U12(x): 2x + 1
- __(x,y): y + x
- active(x): x
- active#(x): 2x
- isNePal(x): 2x + 1
- mark(x): x
- mark#(x): 2x
- nil: 0
- tt: 0
Improved Usable rules
| isNePal(active(X)) | → | isNePal(X) | | active(__(nil, X)) | → | mark(X) |
| U11(mark(X)) | → | U11(X) | | active(__(X, nil)) | → | mark(X) |
| __(active(X1), X2) | → | __(X1, X2) | | mark(__(X1, X2)) | → | active(__(mark(X1), mark(X2))) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| active(U12(tt)) | → | mark(tt) | | __(X1, mark(X2)) | → | __(X1, X2) |
| active(U11(tt)) | → | mark(U12(tt)) | | mark(U11(X)) | → | active(U11(mark(X))) |
| U11(active(X)) | → | U11(X) | | mark(isNePal(X)) | → | active(isNePal(mark(X))) |
| __(mark(X1), X2) | → | __(X1, X2) | | U12(active(X)) | → | U12(X) |
| active(__(__(X, Y), Z)) | → | mark(__(X, __(Y, Z))) | | isNePal(mark(X)) | → | isNePal(X) |
| active(isNePal(__(I, __(P, I)))) | → | mark(U11(tt)) | | mark(nil) | → | active(nil) |
| __(X1, active(X2)) | → | __(X1, X2) | | U12(mark(X)) | → | U12(X) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| mark#(U12(X)) | → | mark#(X) | | mark#(U11(X)) | → | mark#(X) |
| mark#(isNePal(X)) | → | mark#(X) |
Problem 9: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| active#(U11(tt)) | → | mark#(U12(tt)) | | active#(__(__(X, Y), Z)) | → | mark#(__(X, __(Y, Z))) |
| active#(__(X, nil)) | → | mark#(X) | | active#(__(nil, X)) | → | mark#(X) |
| mark#(U11(X)) | → | active#(U11(mark(X))) | | mark#(__(X1, X2)) | → | mark#(X1) |
| mark#(__(X1, X2)) | → | active#(__(mark(X1), mark(X2))) | | active#(isNePal(__(I, __(P, I)))) | → | mark#(U11(tt)) |
| mark#(isNePal(X)) | → | active#(isNePal(mark(X))) | | mark#(__(X1, X2)) | → | mark#(X2) |
Rewrite Rules
| active(__(__(X, Y), Z)) | → | mark(__(X, __(Y, Z))) | | active(__(X, nil)) | → | mark(X) |
| active(__(nil, X)) | → | mark(X) | | active(U11(tt)) | → | mark(U12(tt)) |
| active(U12(tt)) | → | mark(tt) | | active(isNePal(__(I, __(P, I)))) | → | mark(U11(tt)) |
| mark(__(X1, X2)) | → | active(__(mark(X1), mark(X2))) | | mark(nil) | → | active(nil) |
| mark(U11(X)) | → | active(U11(mark(X))) | | mark(tt) | → | active(tt) |
| mark(U12(X)) | → | active(U12(mark(X))) | | mark(isNePal(X)) | → | active(isNePal(mark(X))) |
| __(mark(X1), X2) | → | __(X1, X2) | | __(X1, mark(X2)) | → | __(X1, X2) |
| __(active(X1), X2) | → | __(X1, X2) | | __(X1, active(X2)) | → | __(X1, X2) |
| U11(mark(X)) | → | U11(X) | | U11(active(X)) | → | U11(X) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNePal(mark(X)) | → | isNePal(X) | | isNePal(active(X)) | → | isNePal(X) |
Original Signature
Termination of terms over the following signature is verified: tt, isNePal, active, __, U11, mark, U12, nil
Strategy
Polynomial Interpretation
- U11(x): x + 1
- U12(x): 0
- __(x,y): y + 2x
- active(x): x
- active#(x): 2x
- isNePal(x): 1
- mark(x): x
- mark#(x): 2x
- nil: 0
- tt: 0
Improved Usable rules
| isNePal(active(X)) | → | isNePal(X) | | active(__(nil, X)) | → | mark(X) |
| U11(mark(X)) | → | U11(X) | | active(__(X, nil)) | → | mark(X) |
| __(active(X1), X2) | → | __(X1, X2) | | mark(__(X1, X2)) | → | active(__(mark(X1), mark(X2))) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| active(U12(tt)) | → | mark(tt) | | __(X1, mark(X2)) | → | __(X1, X2) |
| active(U11(tt)) | → | mark(U12(tt)) | | mark(U11(X)) | → | active(U11(mark(X))) |
| U11(active(X)) | → | U11(X) | | mark(isNePal(X)) | → | active(isNePal(mark(X))) |
| __(mark(X1), X2) | → | __(X1, X2) | | U12(active(X)) | → | U12(X) |
| active(__(__(X, Y), Z)) | → | mark(__(X, __(Y, Z))) | | isNePal(mark(X)) | → | isNePal(X) |
| active(isNePal(__(I, __(P, I)))) | → | mark(U11(tt)) | | mark(nil) | → | active(nil) |
| __(X1, active(X2)) | → | __(X1, X2) | | U12(mark(X)) | → | U12(X) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| active#(U11(tt)) | → | mark#(U12(tt)) |
Problem 10: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| active#(__(__(X, Y), Z)) | → | mark#(__(X, __(Y, Z))) | | active#(__(X, nil)) | → | mark#(X) |
| active#(__(nil, X)) | → | mark#(X) | | mark#(__(X1, X2)) | → | active#(__(mark(X1), mark(X2))) |
| mark#(__(X1, X2)) | → | mark#(X1) | | mark#(U11(X)) | → | active#(U11(mark(X))) |
| mark#(isNePal(X)) | → | active#(isNePal(mark(X))) | | active#(isNePal(__(I, __(P, I)))) | → | mark#(U11(tt)) |
| mark#(__(X1, X2)) | → | mark#(X2) |
Rewrite Rules
| active(__(__(X, Y), Z)) | → | mark(__(X, __(Y, Z))) | | active(__(X, nil)) | → | mark(X) |
| active(__(nil, X)) | → | mark(X) | | active(U11(tt)) | → | mark(U12(tt)) |
| active(U12(tt)) | → | mark(tt) | | active(isNePal(__(I, __(P, I)))) | → | mark(U11(tt)) |
| mark(__(X1, X2)) | → | active(__(mark(X1), mark(X2))) | | mark(nil) | → | active(nil) |
| mark(U11(X)) | → | active(U11(mark(X))) | | mark(tt) | → | active(tt) |
| mark(U12(X)) | → | active(U12(mark(X))) | | mark(isNePal(X)) | → | active(isNePal(mark(X))) |
| __(mark(X1), X2) | → | __(X1, X2) | | __(X1, mark(X2)) | → | __(X1, X2) |
| __(active(X1), X2) | → | __(X1, X2) | | __(X1, active(X2)) | → | __(X1, X2) |
| U11(mark(X)) | → | U11(X) | | U11(active(X)) | → | U11(X) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNePal(mark(X)) | → | isNePal(X) | | isNePal(active(X)) | → | isNePal(X) |
Original Signature
Termination of terms over the following signature is verified: tt, isNePal, U11, __, active, U12, mark, nil
Strategy
Polynomial Interpretation
- U11(x): 0
- U12(x): 1
- __(x,y): 1
- active(x): 1
- active#(x): 2x
- isNePal(x): 1
- mark(x): 1
- mark#(x): 2
- nil: 0
- tt: 3
Improved Usable rules
| isNePal(active(X)) | → | isNePal(X) | | __(mark(X1), X2) | → | __(X1, X2) |
| U11(mark(X)) | → | U11(X) | | isNePal(mark(X)) | → | isNePal(X) |
| __(active(X1), X2) | → | __(X1, X2) | | __(X1, mark(X2)) | → | __(X1, X2) |
| __(X1, active(X2)) | → | __(X1, X2) | | U11(active(X)) | → | U11(X) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| mark#(U11(X)) | → | active#(U11(mark(X))) |
Problem 11: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| active#(__(__(X, Y), Z)) | → | mark#(__(X, __(Y, Z))) | | active#(__(X, nil)) | → | mark#(X) |
| active#(__(nil, X)) | → | mark#(X) | | mark#(__(X1, X2)) | → | mark#(X1) |
| mark#(__(X1, X2)) | → | active#(__(mark(X1), mark(X2))) | | active#(isNePal(__(I, __(P, I)))) | → | mark#(U11(tt)) |
| mark#(isNePal(X)) | → | active#(isNePal(mark(X))) | | mark#(__(X1, X2)) | → | mark#(X2) |
Rewrite Rules
| active(__(__(X, Y), Z)) | → | mark(__(X, __(Y, Z))) | | active(__(X, nil)) | → | mark(X) |
| active(__(nil, X)) | → | mark(X) | | active(U11(tt)) | → | mark(U12(tt)) |
| active(U12(tt)) | → | mark(tt) | | active(isNePal(__(I, __(P, I)))) | → | mark(U11(tt)) |
| mark(__(X1, X2)) | → | active(__(mark(X1), mark(X2))) | | mark(nil) | → | active(nil) |
| mark(U11(X)) | → | active(U11(mark(X))) | | mark(tt) | → | active(tt) |
| mark(U12(X)) | → | active(U12(mark(X))) | | mark(isNePal(X)) | → | active(isNePal(mark(X))) |
| __(mark(X1), X2) | → | __(X1, X2) | | __(X1, mark(X2)) | → | __(X1, X2) |
| __(active(X1), X2) | → | __(X1, X2) | | __(X1, active(X2)) | → | __(X1, X2) |
| U11(mark(X)) | → | U11(X) | | U11(active(X)) | → | U11(X) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNePal(mark(X)) | → | isNePal(X) | | isNePal(active(X)) | → | isNePal(X) |
Original Signature
Termination of terms over the following signature is verified: tt, isNePal, active, __, U11, mark, U12, nil
Strategy
Polynomial Interpretation
- U11(x): 1
- U12(x): 0
- __(x,y): y + x
- active(x): x
- active#(x): 2x
- isNePal(x): 2x + 1
- mark(x): x
- mark#(x): 2x
- nil: 1
- tt: 0
Improved Usable rules
| isNePal(active(X)) | → | isNePal(X) | | active(__(nil, X)) | → | mark(X) |
| U11(mark(X)) | → | U11(X) | | active(__(X, nil)) | → | mark(X) |
| __(active(X1), X2) | → | __(X1, X2) | | mark(__(X1, X2)) | → | active(__(mark(X1), mark(X2))) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| active(U12(tt)) | → | mark(tt) | | __(X1, mark(X2)) | → | __(X1, X2) |
| active(U11(tt)) | → | mark(U12(tt)) | | mark(U11(X)) | → | active(U11(mark(X))) |
| U11(active(X)) | → | U11(X) | | mark(isNePal(X)) | → | active(isNePal(mark(X))) |
| __(mark(X1), X2) | → | __(X1, X2) | | U12(active(X)) | → | U12(X) |
| active(__(__(X, Y), Z)) | → | mark(__(X, __(Y, Z))) | | isNePal(mark(X)) | → | isNePal(X) |
| active(isNePal(__(I, __(P, I)))) | → | mark(U11(tt)) | | mark(nil) | → | active(nil) |
| __(X1, active(X2)) | → | __(X1, X2) | | U12(mark(X)) | → | U12(X) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| active#(__(X, nil)) | → | mark#(X) | | active#(__(nil, X)) | → | mark#(X) |
Problem 12: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| active#(__(__(X, Y), Z)) | → | mark#(__(X, __(Y, Z))) | | mark#(__(X1, X2)) | → | active#(__(mark(X1), mark(X2))) |
| mark#(__(X1, X2)) | → | mark#(X1) | | mark#(isNePal(X)) | → | active#(isNePal(mark(X))) |
| active#(isNePal(__(I, __(P, I)))) | → | mark#(U11(tt)) | | mark#(__(X1, X2)) | → | mark#(X2) |
Rewrite Rules
| active(__(__(X, Y), Z)) | → | mark(__(X, __(Y, Z))) | | active(__(X, nil)) | → | mark(X) |
| active(__(nil, X)) | → | mark(X) | | active(U11(tt)) | → | mark(U12(tt)) |
| active(U12(tt)) | → | mark(tt) | | active(isNePal(__(I, __(P, I)))) | → | mark(U11(tt)) |
| mark(__(X1, X2)) | → | active(__(mark(X1), mark(X2))) | | mark(nil) | → | active(nil) |
| mark(U11(X)) | → | active(U11(mark(X))) | | mark(tt) | → | active(tt) |
| mark(U12(X)) | → | active(U12(mark(X))) | | mark(isNePal(X)) | → | active(isNePal(mark(X))) |
| __(mark(X1), X2) | → | __(X1, X2) | | __(X1, mark(X2)) | → | __(X1, X2) |
| __(active(X1), X2) | → | __(X1, X2) | | __(X1, active(X2)) | → | __(X1, X2) |
| U11(mark(X)) | → | U11(X) | | U11(active(X)) | → | U11(X) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNePal(mark(X)) | → | isNePal(X) | | isNePal(active(X)) | → | isNePal(X) |
Original Signature
Termination of terms over the following signature is verified: tt, isNePal, U11, __, active, U12, mark, nil
Strategy
Polynomial Interpretation
- U11(x): 0
- U12(x): 0
- __(x,y): y + 2x
- active(x): x
- active#(x): x
- isNePal(x): 2
- mark(x): x
- mark#(x): x
- nil: 0
- tt: 0
Improved Usable rules
| isNePal(active(X)) | → | isNePal(X) | | active(__(nil, X)) | → | mark(X) |
| U11(mark(X)) | → | U11(X) | | active(__(X, nil)) | → | mark(X) |
| __(active(X1), X2) | → | __(X1, X2) | | mark(__(X1, X2)) | → | active(__(mark(X1), mark(X2))) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| active(U12(tt)) | → | mark(tt) | | __(X1, mark(X2)) | → | __(X1, X2) |
| active(U11(tt)) | → | mark(U12(tt)) | | mark(U11(X)) | → | active(U11(mark(X))) |
| U11(active(X)) | → | U11(X) | | mark(isNePal(X)) | → | active(isNePal(mark(X))) |
| __(mark(X1), X2) | → | __(X1, X2) | | U12(active(X)) | → | U12(X) |
| active(__(__(X, Y), Z)) | → | mark(__(X, __(Y, Z))) | | isNePal(mark(X)) | → | isNePal(X) |
| active(isNePal(__(I, __(P, I)))) | → | mark(U11(tt)) | | mark(nil) | → | active(nil) |
| __(X1, active(X2)) | → | __(X1, X2) | | U12(mark(X)) | → | U12(X) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| active#(isNePal(__(I, __(P, I)))) | → | mark#(U11(tt)) |
Problem 13: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| active#(__(__(X, Y), Z)) | → | mark#(__(X, __(Y, Z))) | | mark#(__(X1, X2)) | → | mark#(X1) |
| mark#(__(X1, X2)) | → | active#(__(mark(X1), mark(X2))) | | mark#(isNePal(X)) | → | active#(isNePal(mark(X))) |
| mark#(__(X1, X2)) | → | mark#(X2) |
Rewrite Rules
| active(__(__(X, Y), Z)) | → | mark(__(X, __(Y, Z))) | | active(__(X, nil)) | → | mark(X) |
| active(__(nil, X)) | → | mark(X) | | active(U11(tt)) | → | mark(U12(tt)) |
| active(U12(tt)) | → | mark(tt) | | active(isNePal(__(I, __(P, I)))) | → | mark(U11(tt)) |
| mark(__(X1, X2)) | → | active(__(mark(X1), mark(X2))) | | mark(nil) | → | active(nil) |
| mark(U11(X)) | → | active(U11(mark(X))) | | mark(tt) | → | active(tt) |
| mark(U12(X)) | → | active(U12(mark(X))) | | mark(isNePal(X)) | → | active(isNePal(mark(X))) |
| __(mark(X1), X2) | → | __(X1, X2) | | __(X1, mark(X2)) | → | __(X1, X2) |
| __(active(X1), X2) | → | __(X1, X2) | | __(X1, active(X2)) | → | __(X1, X2) |
| U11(mark(X)) | → | U11(X) | | U11(active(X)) | → | U11(X) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNePal(mark(X)) | → | isNePal(X) | | isNePal(active(X)) | → | isNePal(X) |
Original Signature
Termination of terms over the following signature is verified: tt, isNePal, active, __, U11, mark, U12, nil
Strategy
Polynomial Interpretation
- U11(x): 0
- U12(x): 0
- __(x,y): y + x + 1
- active(x): x
- active#(x): x
- isNePal(x): 1
- mark(x): x
- mark#(x): x
- nil: 0
- tt: 0
Improved Usable rules
| isNePal(active(X)) | → | isNePal(X) | | active(__(nil, X)) | → | mark(X) |
| U11(mark(X)) | → | U11(X) | | active(__(X, nil)) | → | mark(X) |
| __(active(X1), X2) | → | __(X1, X2) | | mark(__(X1, X2)) | → | active(__(mark(X1), mark(X2))) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| active(U12(tt)) | → | mark(tt) | | __(X1, mark(X2)) | → | __(X1, X2) |
| active(U11(tt)) | → | mark(U12(tt)) | | mark(U11(X)) | → | active(U11(mark(X))) |
| U11(active(X)) | → | U11(X) | | mark(isNePal(X)) | → | active(isNePal(mark(X))) |
| __(mark(X1), X2) | → | __(X1, X2) | | U12(active(X)) | → | U12(X) |
| active(__(__(X, Y), Z)) | → | mark(__(X, __(Y, Z))) | | isNePal(mark(X)) | → | isNePal(X) |
| active(isNePal(__(I, __(P, I)))) | → | mark(U11(tt)) | | mark(nil) | → | active(nil) |
| __(X1, active(X2)) | → | __(X1, X2) | | U12(mark(X)) | → | U12(X) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| mark#(__(X1, X2)) | → | mark#(X1) | | mark#(__(X1, X2)) | → | mark#(X2) |
Problem 14: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| active#(__(__(X, Y), Z)) | → | mark#(__(X, __(Y, Z))) | | mark#(__(X1, X2)) | → | active#(__(mark(X1), mark(X2))) |
| mark#(isNePal(X)) | → | active#(isNePal(mark(X))) |
Rewrite Rules
| active(__(__(X, Y), Z)) | → | mark(__(X, __(Y, Z))) | | active(__(X, nil)) | → | mark(X) |
| active(__(nil, X)) | → | mark(X) | | active(U11(tt)) | → | mark(U12(tt)) |
| active(U12(tt)) | → | mark(tt) | | active(isNePal(__(I, __(P, I)))) | → | mark(U11(tt)) |
| mark(__(X1, X2)) | → | active(__(mark(X1), mark(X2))) | | mark(nil) | → | active(nil) |
| mark(U11(X)) | → | active(U11(mark(X))) | | mark(tt) | → | active(tt) |
| mark(U12(X)) | → | active(U12(mark(X))) | | mark(isNePal(X)) | → | active(isNePal(mark(X))) |
| __(mark(X1), X2) | → | __(X1, X2) | | __(X1, mark(X2)) | → | __(X1, X2) |
| __(active(X1), X2) | → | __(X1, X2) | | __(X1, active(X2)) | → | __(X1, X2) |
| U11(mark(X)) | → | U11(X) | | U11(active(X)) | → | U11(X) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNePal(mark(X)) | → | isNePal(X) | | isNePal(active(X)) | → | isNePal(X) |
Original Signature
Termination of terms over the following signature is verified: tt, isNePal, U11, __, active, U12, mark, nil
Strategy
Polynomial Interpretation
- U11(x): 0
- U12(x): 0
- __(x,y): y + 2x + 1
- active(x): x
- active#(x): x
- isNePal(x): 0
- mark(x): x
- mark#(x): x
- nil: 1
- tt: 0
Improved Usable rules
| isNePal(active(X)) | → | isNePal(X) | | active(__(nil, X)) | → | mark(X) |
| U11(mark(X)) | → | U11(X) | | active(__(X, nil)) | → | mark(X) |
| __(active(X1), X2) | → | __(X1, X2) | | mark(__(X1, X2)) | → | active(__(mark(X1), mark(X2))) |
| mark(tt) | → | active(tt) | | mark(U12(X)) | → | active(U12(mark(X))) |
| active(U12(tt)) | → | mark(tt) | | __(X1, mark(X2)) | → | __(X1, X2) |
| active(U11(tt)) | → | mark(U12(tt)) | | mark(U11(X)) | → | active(U11(mark(X))) |
| U11(active(X)) | → | U11(X) | | mark(isNePal(X)) | → | active(isNePal(mark(X))) |
| __(mark(X1), X2) | → | __(X1, X2) | | U12(active(X)) | → | U12(X) |
| active(__(__(X, Y), Z)) | → | mark(__(X, __(Y, Z))) | | isNePal(mark(X)) | → | isNePal(X) |
| active(isNePal(__(I, __(P, I)))) | → | mark(U11(tt)) | | mark(nil) | → | active(nil) |
| __(X1, active(X2)) | → | __(X1, X2) | | U12(mark(X)) | → | U12(X) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| active#(__(__(X, Y), Z)) | → | mark#(__(X, __(Y, Z))) |
Problem 15: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| mark#(__(X1, X2)) | → | active#(__(mark(X1), mark(X2))) | | mark#(isNePal(X)) | → | active#(isNePal(mark(X))) |
Rewrite Rules
| active(__(__(X, Y), Z)) | → | mark(__(X, __(Y, Z))) | | active(__(X, nil)) | → | mark(X) |
| active(__(nil, X)) | → | mark(X) | | active(U11(tt)) | → | mark(U12(tt)) |
| active(U12(tt)) | → | mark(tt) | | active(isNePal(__(I, __(P, I)))) | → | mark(U11(tt)) |
| mark(__(X1, X2)) | → | active(__(mark(X1), mark(X2))) | | mark(nil) | → | active(nil) |
| mark(U11(X)) | → | active(U11(mark(X))) | | mark(tt) | → | active(tt) |
| mark(U12(X)) | → | active(U12(mark(X))) | | mark(isNePal(X)) | → | active(isNePal(mark(X))) |
| __(mark(X1), X2) | → | __(X1, X2) | | __(X1, mark(X2)) | → | __(X1, X2) |
| __(active(X1), X2) | → | __(X1, X2) | | __(X1, active(X2)) | → | __(X1, X2) |
| U11(mark(X)) | → | U11(X) | | U11(active(X)) | → | U11(X) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNePal(mark(X)) | → | isNePal(X) | | isNePal(active(X)) | → | isNePal(X) |
Original Signature
Termination of terms over the following signature is verified: tt, isNePal, active, __, U11, mark, U12, nil
Strategy
There are no SCCs!
Problem 4: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| isNePal#(active(X)) | → | isNePal#(X) | | isNePal#(mark(X)) | → | isNePal#(X) |
Rewrite Rules
| active(__(__(X, Y), Z)) | → | mark(__(X, __(Y, Z))) | | active(__(X, nil)) | → | mark(X) |
| active(__(nil, X)) | → | mark(X) | | active(U11(tt)) | → | mark(U12(tt)) |
| active(U12(tt)) | → | mark(tt) | | active(isNePal(__(I, __(P, I)))) | → | mark(U11(tt)) |
| mark(__(X1, X2)) | → | active(__(mark(X1), mark(X2))) | | mark(nil) | → | active(nil) |
| mark(U11(X)) | → | active(U11(mark(X))) | | mark(tt) | → | active(tt) |
| mark(U12(X)) | → | active(U12(mark(X))) | | mark(isNePal(X)) | → | active(isNePal(mark(X))) |
| __(mark(X1), X2) | → | __(X1, X2) | | __(X1, mark(X2)) | → | __(X1, X2) |
| __(active(X1), X2) | → | __(X1, X2) | | __(X1, active(X2)) | → | __(X1, X2) |
| U11(mark(X)) | → | U11(X) | | U11(active(X)) | → | U11(X) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNePal(mark(X)) | → | isNePal(X) | | isNePal(active(X)) | → | isNePal(X) |
Original Signature
Termination of terms over the following signature is verified: tt, isNePal, active, __, U11, mark, U12, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| isNePal#(active(X)) | → | isNePal#(X) | | isNePal#(mark(X)) | → | isNePal#(X) |
Problem 5: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U11#(active(X)) | → | U11#(X) | | U11#(mark(X)) | → | U11#(X) |
Rewrite Rules
| active(__(__(X, Y), Z)) | → | mark(__(X, __(Y, Z))) | | active(__(X, nil)) | → | mark(X) |
| active(__(nil, X)) | → | mark(X) | | active(U11(tt)) | → | mark(U12(tt)) |
| active(U12(tt)) | → | mark(tt) | | active(isNePal(__(I, __(P, I)))) | → | mark(U11(tt)) |
| mark(__(X1, X2)) | → | active(__(mark(X1), mark(X2))) | | mark(nil) | → | active(nil) |
| mark(U11(X)) | → | active(U11(mark(X))) | | mark(tt) | → | active(tt) |
| mark(U12(X)) | → | active(U12(mark(X))) | | mark(isNePal(X)) | → | active(isNePal(mark(X))) |
| __(mark(X1), X2) | → | __(X1, X2) | | __(X1, mark(X2)) | → | __(X1, X2) |
| __(active(X1), X2) | → | __(X1, X2) | | __(X1, active(X2)) | → | __(X1, X2) |
| U11(mark(X)) | → | U11(X) | | U11(active(X)) | → | U11(X) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNePal(mark(X)) | → | isNePal(X) | | isNePal(active(X)) | → | isNePal(X) |
Original Signature
Termination of terms over the following signature is verified: tt, isNePal, active, __, U11, mark, U12, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U11#(active(X)) | → | U11#(X) | | U11#(mark(X)) | → | U11#(X) |
Problem 6: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| __#(active(X1), X2) | → | __#(X1, X2) | | __#(mark(X1), X2) | → | __#(X1, X2) |
| __#(X1, mark(X2)) | → | __#(X1, X2) | | __#(X1, active(X2)) | → | __#(X1, X2) |
Rewrite Rules
| active(__(__(X, Y), Z)) | → | mark(__(X, __(Y, Z))) | | active(__(X, nil)) | → | mark(X) |
| active(__(nil, X)) | → | mark(X) | | active(U11(tt)) | → | mark(U12(tt)) |
| active(U12(tt)) | → | mark(tt) | | active(isNePal(__(I, __(P, I)))) | → | mark(U11(tt)) |
| mark(__(X1, X2)) | → | active(__(mark(X1), mark(X2))) | | mark(nil) | → | active(nil) |
| mark(U11(X)) | → | active(U11(mark(X))) | | mark(tt) | → | active(tt) |
| mark(U12(X)) | → | active(U12(mark(X))) | | mark(isNePal(X)) | → | active(isNePal(mark(X))) |
| __(mark(X1), X2) | → | __(X1, X2) | | __(X1, mark(X2)) | → | __(X1, X2) |
| __(active(X1), X2) | → | __(X1, X2) | | __(X1, active(X2)) | → | __(X1, X2) |
| U11(mark(X)) | → | U11(X) | | U11(active(X)) | → | U11(X) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNePal(mark(X)) | → | isNePal(X) | | isNePal(active(X)) | → | isNePal(X) |
Original Signature
Termination of terms over the following signature is verified: tt, isNePal, active, __, U11, mark, U12, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| __#(mark(X1), X2) | → | __#(X1, X2) | | __#(active(X1), X2) | → | __#(X1, X2) |
Problem 7: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| __#(X1, mark(X2)) | → | __#(X1, X2) | | __#(X1, active(X2)) | → | __#(X1, X2) |
Rewrite Rules
| active(__(__(X, Y), Z)) | → | mark(__(X, __(Y, Z))) | | active(__(X, nil)) | → | mark(X) |
| active(__(nil, X)) | → | mark(X) | | active(U11(tt)) | → | mark(U12(tt)) |
| active(U12(tt)) | → | mark(tt) | | active(isNePal(__(I, __(P, I)))) | → | mark(U11(tt)) |
| mark(__(X1, X2)) | → | active(__(mark(X1), mark(X2))) | | mark(nil) | → | active(nil) |
| mark(U11(X)) | → | active(U11(mark(X))) | | mark(tt) | → | active(tt) |
| mark(U12(X)) | → | active(U12(mark(X))) | | mark(isNePal(X)) | → | active(isNePal(mark(X))) |
| __(mark(X1), X2) | → | __(X1, X2) | | __(X1, mark(X2)) | → | __(X1, X2) |
| __(active(X1), X2) | → | __(X1, X2) | | __(X1, active(X2)) | → | __(X1, X2) |
| U11(mark(X)) | → | U11(X) | | U11(active(X)) | → | U11(X) |
| U12(mark(X)) | → | U12(X) | | U12(active(X)) | → | U12(X) |
| isNePal(mark(X)) | → | isNePal(X) | | isNePal(active(X)) | → | isNePal(X) |
Original Signature
Termination of terms over the following signature is verified: tt, isNePal, U11, __, active, U12, mark, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| __#(X1, mark(X2)) | → | __#(X1, X2) | | __#(X1, active(X2)) | → | __#(X1, X2) |