YES
The TRS could be proven terminating. The proof took 41693 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (344ms).
| – Problem 2 was processed with processor PolynomialLinearRange4 (176ms).
| | – Problem 9 was processed with processor PolynomialLinearRange4 (148ms).
| | | – Problem 10 was processed with processor PolynomialLinearRange4 (295ms).
| | | | – Problem 11 was processed with processor PolynomialLinearRange4 (110ms).
| | | | | – Problem 12 was processed with processor PolynomialLinearRange4 (112ms).
| | | | | | – Problem 13 was processed with processor DependencyGraph (1ms).
| | | | | | | – Problem 14 was processed with processor PolynomialLinearRange4 (31ms).
| | | | | | | | – Problem 15 was processed with processor PolynomialLinearRange4 (42ms).
| – Problem 3 was processed with processor SubtermCriterion (0ms).
| – Problem 4 was processed with processor SubtermCriterion (34ms).
| – Problem 5 was processed with processor SubtermCriterion (0ms).
| | – Problem 7 was processed with processor SubtermCriterion (1ms).
| – Problem 6 was processed with processor SubtermCriterion (1ms).
| | – Problem 8 was processed with processor SubtermCriterion (0ms).
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| mark#(cons(X1, X2)) | → | active#(cons(mark(X1), X2)) | | active#(first(0, X)) | → | mark#(nil) |
| cons#(mark(X1), X2) | → | cons#(X1, X2) | | from#(mark(X)) | → | from#(X) |
| mark#(s(X)) | → | s#(mark(X)) | | active#(first(s(X), cons(Y, Z))) | → | cons#(Y, first(X, Z)) |
| active#(first(s(X), cons(Y, Z))) | → | first#(X, Z) | | first#(X1, active(X2)) | → | first#(X1, X2) |
| mark#(nil) | → | active#(nil) | | active#(first(s(X), cons(Y, Z))) | → | mark#(cons(Y, first(X, Z))) |
| active#(from(X)) | → | cons#(X, from(s(X))) | | mark#(s(X)) | → | mark#(X) |
| cons#(X1, mark(X2)) | → | cons#(X1, X2) | | first#(mark(X1), X2) | → | first#(X1, X2) |
| mark#(first(X1, X2)) | → | first#(mark(X1), mark(X2)) | | mark#(cons(X1, X2)) | → | cons#(mark(X1), X2) |
| first#(X1, mark(X2)) | → | first#(X1, X2) | | mark#(0) | → | active#(0) |
| mark#(first(X1, X2)) | → | mark#(X2) | | mark#(s(X)) | → | active#(s(mark(X))) |
| mark#(from(X)) | → | mark#(X) | | mark#(first(X1, X2)) | → | active#(first(mark(X1), mark(X2))) |
| mark#(cons(X1, X2)) | → | mark#(X1) | | cons#(active(X1), X2) | → | cons#(X1, X2) |
| active#(from(X)) | → | mark#(cons(X, from(s(X)))) | | active#(from(X)) | → | s#(X) |
| s#(mark(X)) | → | s#(X) | | cons#(X1, active(X2)) | → | cons#(X1, X2) |
| mark#(from(X)) | → | from#(mark(X)) | | mark#(from(X)) | → | active#(from(mark(X))) |
| first#(active(X1), X2) | → | first#(X1, X2) | | from#(active(X)) | → | from#(X) |
| mark#(first(X1, X2)) | → | mark#(X1) | | s#(active(X)) | → | s#(X) |
| active#(from(X)) | → | from#(s(X)) |
Rewrite Rules
| active(first(0, X)) | → | mark(nil) | | active(first(s(X), cons(Y, Z))) | → | mark(cons(Y, first(X, Z))) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | mark(first(X1, X2)) | → | active(first(mark(X1), mark(X2))) |
| mark(0) | → | active(0) | | mark(nil) | → | active(nil) |
| mark(s(X)) | → | active(s(mark(X))) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(from(X)) | → | active(from(mark(X))) | | first(mark(X1), X2) | → | first(X1, X2) |
| first(X1, mark(X2)) | → | first(X1, X2) | | first(active(X1), X2) | → | first(X1, X2) |
| first(X1, active(X2)) | → | first(X1, X2) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | 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) |
Original Signature
Termination of terms over the following signature is verified: 0, s, active, mark, from, first, nil, cons
Strategy
The following SCCs where found
| mark#(first(X1, X2)) → mark#(X2) | mark#(cons(X1, X2)) → active#(cons(mark(X1), X2)) |
| mark#(s(X)) → active#(s(mark(X))) | mark#(from(X)) → active#(from(mark(X))) |
| active#(first(s(X), cons(Y, Z))) → mark#(cons(Y, first(X, Z))) | mark#(from(X)) → mark#(X) |
| mark#(first(X1, X2)) → active#(first(mark(X1), mark(X2))) | mark#(cons(X1, X2)) → mark#(X1) |
| mark#(first(X1, X2)) → mark#(X1) | mark#(s(X)) → mark#(X) |
| active#(from(X)) → mark#(cons(X, from(s(X)))) |
| from#(active(X)) → from#(X) | from#(mark(X)) → from#(X) |
| s#(mark(X)) → s#(X) | s#(active(X)) → s#(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) |
| first#(active(X1), X2) → first#(X1, X2) | first#(mark(X1), X2) → first#(X1, X2) |
| first#(X1, mark(X2)) → first#(X1, X2) | first#(X1, active(X2)) → first#(X1, X2) |
Problem 2: PolynomialLinearRange4
Dependency Pair Problem
Dependency Pairs
| mark#(first(X1, X2)) | → | mark#(X2) | | mark#(cons(X1, X2)) | → | active#(cons(mark(X1), X2)) |
| mark#(s(X)) | → | active#(s(mark(X))) | | mark#(from(X)) | → | active#(from(mark(X))) |
| active#(first(s(X), cons(Y, Z))) | → | mark#(cons(Y, first(X, Z))) | | mark#(from(X)) | → | mark#(X) |
| mark#(first(X1, X2)) | → | active#(first(mark(X1), mark(X2))) | | mark#(cons(X1, X2)) | → | mark#(X1) |
| mark#(first(X1, X2)) | → | mark#(X1) | | mark#(s(X)) | → | mark#(X) |
| active#(from(X)) | → | mark#(cons(X, from(s(X)))) |
Rewrite Rules
| active(first(0, X)) | → | mark(nil) | | active(first(s(X), cons(Y, Z))) | → | mark(cons(Y, first(X, Z))) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | mark(first(X1, X2)) | → | active(first(mark(X1), mark(X2))) |
| mark(0) | → | active(0) | | mark(nil) | → | active(nil) |
| mark(s(X)) | → | active(s(mark(X))) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(from(X)) | → | active(from(mark(X))) | | first(mark(X1), X2) | → | first(X1, X2) |
| first(X1, mark(X2)) | → | first(X1, X2) | | first(active(X1), X2) | → | first(X1, X2) |
| first(X1, active(X2)) | → | first(X1, X2) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | 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) |
Original Signature
Termination of terms over the following signature is verified: 0, s, active, mark, from, first, nil, cons
Strategy
Polynomial Interpretation
- 0: 2
- active(x): 0
- active#(x): 2x
- cons(x,y): 1
- first(x,y): 1
- from(x): 1
- mark(x): 0
- mark#(x): 2
- nil: 1
- s(x): 0
Standard Usable rules
| cons(active(X1), X2) | → | cons(X1, X2) | | from(mark(X)) | → | from(X) |
| mark(first(X1, X2)) | → | active(first(mark(X1), mark(X2))) | | mark(s(X)) | → | active(s(mark(X))) |
| first(mark(X1), X2) | → | first(X1, X2) | | active(first(0, X)) | → | mark(nil) |
| active(first(s(X), cons(Y, Z))) | → | mark(cons(Y, first(X, Z))) | | cons(X1, mark(X2)) | → | cons(X1, X2) |
| first(X1, mark(X2)) | → | first(X1, X2) | | first(active(X1), X2) | → | first(X1, X2) |
| first(X1, active(X2)) | → | first(X1, X2) | | cons(mark(X1), X2) | → | cons(X1, X2) |
| mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) | | mark(from(X)) | → | active(from(mark(X))) |
| s(mark(X)) | → | s(X) | | active(from(X)) | → | mark(cons(X, from(s(X)))) |
| mark(nil) | → | active(nil) | | mark(0) | → | active(0) |
| s(active(X)) | → | s(X) | | from(active(X)) | → | from(X) |
| cons(X1, active(X2)) | → | cons(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 9: PolynomialLinearRange4
Dependency Pair Problem
Dependency Pairs
| mark#(cons(X1, X2)) | → | active#(cons(mark(X1), X2)) | | mark#(first(X1, X2)) | → | mark#(X2) |
| mark#(from(X)) | → | active#(from(mark(X))) | | active#(first(s(X), cons(Y, Z))) | → | mark#(cons(Y, first(X, Z))) |
| mark#(from(X)) | → | mark#(X) | | mark#(cons(X1, X2)) | → | mark#(X1) |
| mark#(first(X1, X2)) | → | active#(first(mark(X1), mark(X2))) | | mark#(first(X1, X2)) | → | mark#(X1) |
| mark#(s(X)) | → | mark#(X) | | active#(from(X)) | → | mark#(cons(X, from(s(X)))) |
Rewrite Rules
| active(first(0, X)) | → | mark(nil) | | active(first(s(X), cons(Y, Z))) | → | mark(cons(Y, first(X, Z))) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | mark(first(X1, X2)) | → | active(first(mark(X1), mark(X2))) |
| mark(0) | → | active(0) | | mark(nil) | → | active(nil) |
| mark(s(X)) | → | active(s(mark(X))) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(from(X)) | → | active(from(mark(X))) | | first(mark(X1), X2) | → | first(X1, X2) |
| first(X1, mark(X2)) | → | first(X1, X2) | | first(active(X1), X2) | → | first(X1, X2) |
| first(X1, active(X2)) | → | first(X1, X2) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | 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) |
Original Signature
Termination of terms over the following signature is verified: 0, s, active, mark, from, first, cons, nil
Strategy
Polynomial Interpretation
- 0: 3
- active(x): x
- active#(x): 2x + 1
- cons(x,y): x
- first(x,y): 2y + x
- from(x): x + 1
- mark(x): x
- mark#(x): 2x + 1
- nil: 3
- s(x): x
Standard Usable rules
| cons(active(X1), X2) | → | cons(X1, X2) | | from(mark(X)) | → | from(X) |
| mark(first(X1, X2)) | → | active(first(mark(X1), mark(X2))) | | mark(s(X)) | → | active(s(mark(X))) |
| first(mark(X1), X2) | → | first(X1, X2) | | active(first(0, X)) | → | mark(nil) |
| active(first(s(X), cons(Y, Z))) | → | mark(cons(Y, first(X, Z))) | | cons(X1, mark(X2)) | → | cons(X1, X2) |
| first(X1, mark(X2)) | → | first(X1, X2) | | first(active(X1), X2) | → | first(X1, X2) |
| first(X1, active(X2)) | → | first(X1, X2) | | cons(mark(X1), X2) | → | cons(X1, X2) |
| mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) | | mark(from(X)) | → | active(from(mark(X))) |
| s(mark(X)) | → | s(X) | | active(from(X)) | → | mark(cons(X, from(s(X)))) |
| mark(nil) | → | active(nil) | | mark(0) | → | active(0) |
| s(active(X)) | → | s(X) | | from(active(X)) | → | from(X) |
| cons(X1, active(X2)) | → | cons(X1, X2) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| mark#(from(X)) | → | mark#(X) | | active#(from(X)) | → | mark#(cons(X, from(s(X)))) |
Problem 10: PolynomialLinearRange4
Dependency Pair Problem
Dependency Pairs
| mark#(first(X1, X2)) | → | mark#(X2) | | mark#(cons(X1, X2)) | → | active#(cons(mark(X1), X2)) |
| mark#(from(X)) | → | active#(from(mark(X))) | | active#(first(s(X), cons(Y, Z))) | → | mark#(cons(Y, first(X, Z))) |
| mark#(first(X1, X2)) | → | active#(first(mark(X1), mark(X2))) | | mark#(cons(X1, X2)) | → | mark#(X1) |
| mark#(first(X1, X2)) | → | mark#(X1) | | mark#(s(X)) | → | mark#(X) |
Rewrite Rules
| active(first(0, X)) | → | mark(nil) | | active(first(s(X), cons(Y, Z))) | → | mark(cons(Y, first(X, Z))) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | mark(first(X1, X2)) | → | active(first(mark(X1), mark(X2))) |
| mark(0) | → | active(0) | | mark(nil) | → | active(nil) |
| mark(s(X)) | → | active(s(mark(X))) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(from(X)) | → | active(from(mark(X))) | | first(mark(X1), X2) | → | first(X1, X2) |
| first(X1, mark(X2)) | → | first(X1, X2) | | first(active(X1), X2) | → | first(X1, X2) |
| first(X1, active(X2)) | → | first(X1, X2) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | 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) |
Original Signature
Termination of terms over the following signature is verified: 0, s, active, mark, from, first, nil, cons
Strategy
Polynomial Interpretation
- 0: 3
- active(x): 0
- active#(x): x + 1
- cons(x,y): 2
- first(x,y): 2
- from(x): 0
- mark(x): 0
- mark#(x): 3
- nil: 1
- s(x): 1
Standard Usable rules
| cons(active(X1), X2) | → | cons(X1, X2) | | from(mark(X)) | → | from(X) |
| mark(first(X1, X2)) | → | active(first(mark(X1), mark(X2))) | | mark(s(X)) | → | active(s(mark(X))) |
| first(mark(X1), X2) | → | first(X1, X2) | | active(first(0, X)) | → | mark(nil) |
| active(first(s(X), cons(Y, Z))) | → | mark(cons(Y, first(X, Z))) | | first(X1, mark(X2)) | → | first(X1, X2) |
| cons(X1, mark(X2)) | → | cons(X1, X2) | | first(active(X1), X2) | → | first(X1, X2) |
| first(X1, active(X2)) | → | first(X1, X2) | | cons(mark(X1), X2) | → | cons(X1, X2) |
| mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) | | mark(from(X)) | → | active(from(mark(X))) |
| s(mark(X)) | → | s(X) | | active(from(X)) | → | mark(cons(X, from(s(X)))) |
| mark(nil) | → | active(nil) | | s(active(X)) | → | s(X) |
| mark(0) | → | active(0) | | from(active(X)) | → | from(X) |
| cons(X1, active(X2)) | → | cons(X1, X2) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| mark#(from(X)) | → | active#(from(mark(X))) |
Problem 11: PolynomialLinearRange4
Dependency Pair Problem
Dependency Pairs
| mark#(cons(X1, X2)) | → | active#(cons(mark(X1), X2)) | | mark#(first(X1, X2)) | → | mark#(X2) |
| active#(first(s(X), cons(Y, Z))) | → | mark#(cons(Y, first(X, Z))) | | mark#(cons(X1, X2)) | → | mark#(X1) |
| mark#(first(X1, X2)) | → | active#(first(mark(X1), mark(X2))) | | mark#(first(X1, X2)) | → | mark#(X1) |
| mark#(s(X)) | → | mark#(X) |
Rewrite Rules
| active(first(0, X)) | → | mark(nil) | | active(first(s(X), cons(Y, Z))) | → | mark(cons(Y, first(X, Z))) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | mark(first(X1, X2)) | → | active(first(mark(X1), mark(X2))) |
| mark(0) | → | active(0) | | mark(nil) | → | active(nil) |
| mark(s(X)) | → | active(s(mark(X))) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(from(X)) | → | active(from(mark(X))) | | first(mark(X1), X2) | → | first(X1, X2) |
| first(X1, mark(X2)) | → | first(X1, X2) | | first(active(X1), X2) | → | first(X1, X2) |
| first(X1, active(X2)) | → | first(X1, X2) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | 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) |
Original Signature
Termination of terms over the following signature is verified: 0, s, active, mark, from, first, cons, nil
Strategy
Polynomial Interpretation
- 0: 3
- active(x): 0
- active#(x): x + 1
- cons(x,y): 0
- first(x,y): 1
- from(x): 1
- mark(x): 0
- mark#(x): 2
- nil: 2
- s(x): 0
Standard Usable rules
| cons(active(X1), X2) | → | cons(X1, X2) | | from(mark(X)) | → | from(X) |
| mark(first(X1, X2)) | → | active(first(mark(X1), mark(X2))) | | mark(s(X)) | → | active(s(mark(X))) |
| first(mark(X1), X2) | → | first(X1, X2) | | active(first(0, X)) | → | mark(nil) |
| active(first(s(X), cons(Y, Z))) | → | mark(cons(Y, first(X, Z))) | | first(X1, mark(X2)) | → | first(X1, X2) |
| cons(X1, mark(X2)) | → | cons(X1, X2) | | first(active(X1), X2) | → | first(X1, X2) |
| first(X1, active(X2)) | → | first(X1, X2) | | cons(mark(X1), X2) | → | cons(X1, X2) |
| mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) | | mark(from(X)) | → | active(from(mark(X))) |
| s(mark(X)) | → | s(X) | | active(from(X)) | → | mark(cons(X, from(s(X)))) |
| mark(nil) | → | active(nil) | | s(active(X)) | → | s(X) |
| mark(0) | → | active(0) | | from(active(X)) | → | from(X) |
| cons(X1, active(X2)) | → | cons(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#(first(X1, X2)) | → | mark#(X2) | | active#(first(s(X), cons(Y, Z))) | → | mark#(cons(Y, first(X, Z))) |
| mark#(first(X1, X2)) | → | active#(first(mark(X1), mark(X2))) | | mark#(cons(X1, X2)) | → | mark#(X1) |
| mark#(first(X1, X2)) | → | mark#(X1) | | mark#(s(X)) | → | mark#(X) |
Rewrite Rules
| active(first(0, X)) | → | mark(nil) | | active(first(s(X), cons(Y, Z))) | → | mark(cons(Y, first(X, Z))) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | mark(first(X1, X2)) | → | active(first(mark(X1), mark(X2))) |
| mark(0) | → | active(0) | | mark(nil) | → | active(nil) |
| mark(s(X)) | → | active(s(mark(X))) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(from(X)) | → | active(from(mark(X))) | | first(mark(X1), X2) | → | first(X1, X2) |
| first(X1, mark(X2)) | → | first(X1, X2) | | first(active(X1), X2) | → | first(X1, X2) |
| first(X1, active(X2)) | → | first(X1, X2) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | 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) |
Original Signature
Termination of terms over the following signature is verified: 0, s, active, mark, from, first, nil, cons
Strategy
Polynomial Interpretation
- 0: 0
- active(x): x
- active#(x): x
- cons(x,y): x
- first(x,y): 2y + x
- from(x): 3x
- mark(x): 2x
- mark#(x): 2x
- nil: 0
- s(x): 2x + 1
Standard Usable rules
| cons(active(X1), X2) | → | cons(X1, X2) | | from(mark(X)) | → | from(X) |
| mark(first(X1, X2)) | → | active(first(mark(X1), mark(X2))) | | mark(s(X)) | → | active(s(mark(X))) |
| first(mark(X1), X2) | → | first(X1, X2) | | active(first(0, X)) | → | mark(nil) |
| active(first(s(X), cons(Y, Z))) | → | mark(cons(Y, first(X, Z))) | | first(X1, mark(X2)) | → | first(X1, X2) |
| cons(X1, mark(X2)) | → | cons(X1, X2) | | first(active(X1), X2) | → | first(X1, X2) |
| first(X1, active(X2)) | → | first(X1, X2) | | cons(mark(X1), X2) | → | cons(X1, X2) |
| mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) | | mark(from(X)) | → | active(from(mark(X))) |
| s(mark(X)) | → | s(X) | | active(from(X)) | → | mark(cons(X, from(s(X)))) |
| mark(nil) | → | active(nil) | | s(active(X)) | → | s(X) |
| mark(0) | → | active(0) | | from(active(X)) | → | from(X) |
| cons(X1, active(X2)) | → | cons(X1, X2) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| active#(first(s(X), cons(Y, Z))) | → | mark#(cons(Y, first(X, Z))) | | mark#(s(X)) | → | mark#(X) |
Problem 13: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| mark#(first(X1, X2)) | → | mark#(X2) | | mark#(cons(X1, X2)) | → | mark#(X1) |
| mark#(first(X1, X2)) | → | active#(first(mark(X1), mark(X2))) | | mark#(first(X1, X2)) | → | mark#(X1) |
Rewrite Rules
| active(first(0, X)) | → | mark(nil) | | active(first(s(X), cons(Y, Z))) | → | mark(cons(Y, first(X, Z))) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | mark(first(X1, X2)) | → | active(first(mark(X1), mark(X2))) |
| mark(0) | → | active(0) | | mark(nil) | → | active(nil) |
| mark(s(X)) | → | active(s(mark(X))) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(from(X)) | → | active(from(mark(X))) | | first(mark(X1), X2) | → | first(X1, X2) |
| first(X1, mark(X2)) | → | first(X1, X2) | | first(active(X1), X2) | → | first(X1, X2) |
| first(X1, active(X2)) | → | first(X1, X2) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | 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) |
Original Signature
Termination of terms over the following signature is verified: 0, s, active, mark, from, first, cons, nil
Strategy
The following SCCs where found
| mark#(first(X1, X2)) → mark#(X2) | mark#(cons(X1, X2)) → mark#(X1) |
| mark#(first(X1, X2)) → mark#(X1) |
Problem 14: PolynomialLinearRange4
Dependency Pair Problem
Dependency Pairs
| mark#(first(X1, X2)) | → | mark#(X2) | | mark#(cons(X1, X2)) | → | mark#(X1) |
| mark#(first(X1, X2)) | → | mark#(X1) |
Rewrite Rules
| active(first(0, X)) | → | mark(nil) | | active(first(s(X), cons(Y, Z))) | → | mark(cons(Y, first(X, Z))) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | mark(first(X1, X2)) | → | active(first(mark(X1), mark(X2))) |
| mark(0) | → | active(0) | | mark(nil) | → | active(nil) |
| mark(s(X)) | → | active(s(mark(X))) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(from(X)) | → | active(from(mark(X))) | | first(mark(X1), X2) | → | first(X1, X2) |
| first(X1, mark(X2)) | → | first(X1, X2) | | first(active(X1), X2) | → | first(X1, X2) |
| first(X1, active(X2)) | → | first(X1, X2) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | 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) |
Original Signature
Termination of terms over the following signature is verified: 0, s, active, mark, from, first, cons, nil
Strategy
Polynomial Interpretation
- 0: 0
- active(x): 0
- cons(x,y): x
- first(x,y): y + x + 1
- from(x): 0
- mark(x): 0
- mark#(x): 2x
- nil: 0
- s(x): 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#(first(X1, X2)) | → | mark#(X2) | | mark#(first(X1, X2)) | → | mark#(X1) |
Problem 15: PolynomialLinearRange4
Dependency Pair Problem
Dependency Pairs
| mark#(cons(X1, X2)) | → | mark#(X1) |
Rewrite Rules
| active(first(0, X)) | → | mark(nil) | | active(first(s(X), cons(Y, Z))) | → | mark(cons(Y, first(X, Z))) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | mark(first(X1, X2)) | → | active(first(mark(X1), mark(X2))) |
| mark(0) | → | active(0) | | mark(nil) | → | active(nil) |
| mark(s(X)) | → | active(s(mark(X))) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(from(X)) | → | active(from(mark(X))) | | first(mark(X1), X2) | → | first(X1, X2) |
| first(X1, mark(X2)) | → | first(X1, X2) | | first(active(X1), X2) | → | first(X1, X2) |
| first(X1, active(X2)) | → | first(X1, X2) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | 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) |
Original Signature
Termination of terms over the following signature is verified: 0, s, active, mark, from, first, nil, cons
Strategy
Polynomial Interpretation
- 0: 0
- active(x): 0
- cons(x,y): x + 1
- first(x,y): 0
- from(x): 0
- mark(x): 0
- mark#(x): x + 1
- nil: 0
- s(x): 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 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| from#(active(X)) | → | from#(X) | | from#(mark(X)) | → | from#(X) |
Rewrite Rules
| active(first(0, X)) | → | mark(nil) | | active(first(s(X), cons(Y, Z))) | → | mark(cons(Y, first(X, Z))) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | mark(first(X1, X2)) | → | active(first(mark(X1), mark(X2))) |
| mark(0) | → | active(0) | | mark(nil) | → | active(nil) |
| mark(s(X)) | → | active(s(mark(X))) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(from(X)) | → | active(from(mark(X))) | | first(mark(X1), X2) | → | first(X1, X2) |
| first(X1, mark(X2)) | → | first(X1, X2) | | first(active(X1), X2) | → | first(X1, X2) |
| first(X1, active(X2)) | → | first(X1, X2) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | 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) |
Original Signature
Termination of terms over the following signature is verified: 0, s, active, mark, from, first, 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 4: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| s#(mark(X)) | → | s#(X) | | s#(active(X)) | → | s#(X) |
Rewrite Rules
| active(first(0, X)) | → | mark(nil) | | active(first(s(X), cons(Y, Z))) | → | mark(cons(Y, first(X, Z))) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | mark(first(X1, X2)) | → | active(first(mark(X1), mark(X2))) |
| mark(0) | → | active(0) | | mark(nil) | → | active(nil) |
| mark(s(X)) | → | active(s(mark(X))) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(from(X)) | → | active(from(mark(X))) | | first(mark(X1), X2) | → | first(X1, X2) |
| first(X1, mark(X2)) | → | first(X1, X2) | | first(active(X1), X2) | → | first(X1, X2) |
| first(X1, active(X2)) | → | first(X1, X2) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | 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) |
Original Signature
Termination of terms over the following signature is verified: 0, s, active, mark, from, first, 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 5: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| first#(active(X1), X2) | → | first#(X1, X2) | | first#(mark(X1), X2) | → | first#(X1, X2) |
| first#(X1, mark(X2)) | → | first#(X1, X2) | | first#(X1, active(X2)) | → | first#(X1, X2) |
Rewrite Rules
| active(first(0, X)) | → | mark(nil) | | active(first(s(X), cons(Y, Z))) | → | mark(cons(Y, first(X, Z))) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | mark(first(X1, X2)) | → | active(first(mark(X1), mark(X2))) |
| mark(0) | → | active(0) | | mark(nil) | → | active(nil) |
| mark(s(X)) | → | active(s(mark(X))) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(from(X)) | → | active(from(mark(X))) | | first(mark(X1), X2) | → | first(X1, X2) |
| first(X1, mark(X2)) | → | first(X1, X2) | | first(active(X1), X2) | → | first(X1, X2) |
| first(X1, active(X2)) | → | first(X1, X2) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | 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) |
Original Signature
Termination of terms over the following signature is verified: 0, s, active, mark, from, first, nil, cons
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| first#(active(X1), X2) | → | first#(X1, X2) | | first#(mark(X1), X2) | → | first#(X1, X2) |
Problem 7: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| first#(X1, active(X2)) | → | first#(X1, X2) | | first#(X1, mark(X2)) | → | first#(X1, X2) |
Rewrite Rules
| active(first(0, X)) | → | mark(nil) | | active(first(s(X), cons(Y, Z))) | → | mark(cons(Y, first(X, Z))) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | mark(first(X1, X2)) | → | active(first(mark(X1), mark(X2))) |
| mark(0) | → | active(0) | | mark(nil) | → | active(nil) |
| mark(s(X)) | → | active(s(mark(X))) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(from(X)) | → | active(from(mark(X))) | | first(mark(X1), X2) | → | first(X1, X2) |
| first(X1, mark(X2)) | → | first(X1, X2) | | first(active(X1), X2) | → | first(X1, X2) |
| first(X1, active(X2)) | → | first(X1, X2) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | 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) |
Original Signature
Termination of terms over the following signature is verified: 0, s, active, mark, from, first, cons, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| first#(X1, mark(X2)) | → | first#(X1, X2) | | first#(X1, active(X2)) | → | first#(X1, X2) |
Problem 6: 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(first(0, X)) | → | mark(nil) | | active(first(s(X), cons(Y, Z))) | → | mark(cons(Y, first(X, Z))) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | mark(first(X1, X2)) | → | active(first(mark(X1), mark(X2))) |
| mark(0) | → | active(0) | | mark(nil) | → | active(nil) |
| mark(s(X)) | → | active(s(mark(X))) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(from(X)) | → | active(from(mark(X))) | | first(mark(X1), X2) | → | first(X1, X2) |
| first(X1, mark(X2)) | → | first(X1, X2) | | first(active(X1), X2) | → | first(X1, X2) |
| first(X1, active(X2)) | → | first(X1, X2) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | 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) |
Original Signature
Termination of terms over the following signature is verified: 0, s, active, mark, from, first, 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 8: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| cons#(X1, active(X2)) | → | cons#(X1, X2) | | cons#(X1, mark(X2)) | → | cons#(X1, X2) |
Rewrite Rules
| active(first(0, X)) | → | mark(nil) | | active(first(s(X), cons(Y, Z))) | → | mark(cons(Y, first(X, Z))) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | mark(first(X1, X2)) | → | active(first(mark(X1), mark(X2))) |
| mark(0) | → | active(0) | | mark(nil) | → | active(nil) |
| mark(s(X)) | → | active(s(mark(X))) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(from(X)) | → | active(from(mark(X))) | | first(mark(X1), X2) | → | first(X1, X2) |
| first(X1, mark(X2)) | → | first(X1, X2) | | first(active(X1), X2) | → | first(X1, X2) |
| first(X1, active(X2)) | → | first(X1, X2) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | 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) |
Original Signature
Termination of terms over the following signature is verified: 0, s, active, mark, from, first, 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) |