YES
The TRS could be proven terminating. The proof took 6024 ms.
The following DP Processors were used
Problem 1 was processed with processor PolynomialLinearRange4iUR (2791ms).
| – Problem 2 was processed with processor DependencyGraph (20ms).
| | – Problem 3 was processed with processor PolynomialLinearRange4iUR (1489ms).
| | | – Problem 4 was processed with processor DependencyGraph (23ms).
| | | | – Problem 5 was processed with processor PolynomialLinearRange4iUR (717ms).
| | | | | – Problem 6 was processed with processor DependencyGraph (5ms).
| | | | | | – Problem 7 was processed with processor PolynomialLinearRange4iUR (525ms).
| | | | | | | – Problem 8 was processed with processor PolynomialLinearRange4iUR (7ms).
Problem 1: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| a__sieve#(cons(s(N), Y)) | → | mark#(N) | | mark#(filter(X1, X2, X3)) | → | mark#(X3) |
| mark#(zprimes) | → | a__zprimes# | | mark#(cons(X1, X2)) | → | mark#(X1) |
| a__zprimes# | → | a__sieve#(a__nats(s(s(0)))) | | a__filter#(cons(X, Y), s(N), M) | → | mark#(X) |
| a__nats#(N) | → | mark#(N) | | mark#(sieve(X)) | → | a__sieve#(mark(X)) |
| mark#(nats(X)) | → | a__nats#(mark(X)) | | mark#(filter(X1, X2, X3)) | → | mark#(X1) |
| mark#(filter(X1, X2, X3)) | → | mark#(X2) | | mark#(nats(X)) | → | mark#(X) |
| a__zprimes# | → | a__nats#(s(s(0))) | | mark#(s(X)) | → | mark#(X) |
| mark#(filter(X1, X2, X3)) | → | a__filter#(mark(X1), mark(X2), mark(X3)) | | mark#(sieve(X)) | → | mark#(X) |
Rewrite Rules
| a__filter(cons(X, Y), 0, M) | → | cons(0, filter(Y, M, M)) | | a__filter(cons(X, Y), s(N), M) | → | cons(mark(X), filter(Y, N, M)) |
| a__sieve(cons(0, Y)) | → | cons(0, sieve(Y)) | | a__sieve(cons(s(N), Y)) | → | cons(s(mark(N)), sieve(filter(Y, N, N))) |
| a__nats(N) | → | cons(mark(N), nats(s(N))) | | a__zprimes | → | a__sieve(a__nats(s(s(0)))) |
| mark(filter(X1, X2, X3)) | → | a__filter(mark(X1), mark(X2), mark(X3)) | | mark(sieve(X)) | → | a__sieve(mark(X)) |
| mark(nats(X)) | → | a__nats(mark(X)) | | mark(zprimes) | → | a__zprimes |
| mark(cons(X1, X2)) | → | cons(mark(X1), X2) | | mark(0) | → | 0 |
| mark(s(X)) | → | s(mark(X)) | | a__filter(X1, X2, X3) | → | filter(X1, X2, X3) |
| a__sieve(X) | → | sieve(X) | | a__nats(X) | → | nats(X) |
| a__zprimes | → | zprimes |
Original Signature
Termination of terms over the following signature is verified: a__filter, nats, 0, s, sieve, mark, a__zprimes, zprimes, filter, a__nats, cons, a__sieve
Strategy
Polynomial Interpretation
- 0: 0
- a__filter(x,y,z): z + y + 2x
- a__filter#(x,y,z): x
- a__nats(x): 2x
- a__nats#(x): x
- a__sieve(x): 2x
- a__sieve#(x): x
- a__zprimes: 2
- a__zprimes#: 0
- cons(x,y): x
- filter(x,y,z): z + y + 2x
- mark(x): 2x
- mark#(x): x
- nats(x): 2x
- s(x): 2x
- sieve(x): 2x
- zprimes: 1
Improved Usable rules
| a__nats(N) | → | cons(mark(N), nats(s(N))) | | mark(cons(X1, X2)) | → | cons(mark(X1), X2) |
| a__nats(X) | → | nats(X) | | mark(0) | → | 0 |
| a__sieve(cons(0, Y)) | → | cons(0, sieve(Y)) | | a__zprimes | → | zprimes |
| mark(filter(X1, X2, X3)) | → | a__filter(mark(X1), mark(X2), mark(X3)) | | a__zprimes | → | a__sieve(a__nats(s(s(0)))) |
| a__filter(cons(X, Y), 0, M) | → | cons(0, filter(Y, M, M)) | | a__sieve(cons(s(N), Y)) | → | cons(s(mark(N)), sieve(filter(Y, N, N))) |
| a__filter(cons(X, Y), s(N), M) | → | cons(mark(X), filter(Y, N, M)) | | mark(s(X)) | → | s(mark(X)) |
| mark(sieve(X)) | → | a__sieve(mark(X)) | | mark(zprimes) | → | a__zprimes |
| mark(nats(X)) | → | a__nats(mark(X)) | | a__filter(X1, X2, X3) | → | filter(X1, X2, X3) |
| a__sieve(X) | → | sieve(X) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| mark#(zprimes) | → | a__zprimes# |
Problem 2: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| a__sieve#(cons(s(N), Y)) | → | mark#(N) | | mark#(filter(X1, X2, X3)) | → | mark#(X3) |
| mark#(cons(X1, X2)) | → | mark#(X1) | | a__zprimes# | → | a__sieve#(a__nats(s(s(0)))) |
| a__filter#(cons(X, Y), s(N), M) | → | mark#(X) | | a__nats#(N) | → | mark#(N) |
| mark#(sieve(X)) | → | a__sieve#(mark(X)) | | mark#(nats(X)) | → | a__nats#(mark(X)) |
| mark#(filter(X1, X2, X3)) | → | mark#(X1) | | mark#(filter(X1, X2, X3)) | → | mark#(X2) |
| mark#(nats(X)) | → | mark#(X) | | a__zprimes# | → | a__nats#(s(s(0))) |
| mark#(s(X)) | → | mark#(X) | | mark#(filter(X1, X2, X3)) | → | a__filter#(mark(X1), mark(X2), mark(X3)) |
| mark#(sieve(X)) | → | mark#(X) |
Rewrite Rules
| a__filter(cons(X, Y), 0, M) | → | cons(0, filter(Y, M, M)) | | a__filter(cons(X, Y), s(N), M) | → | cons(mark(X), filter(Y, N, M)) |
| a__sieve(cons(0, Y)) | → | cons(0, sieve(Y)) | | a__sieve(cons(s(N), Y)) | → | cons(s(mark(N)), sieve(filter(Y, N, N))) |
| a__nats(N) | → | cons(mark(N), nats(s(N))) | | a__zprimes | → | a__sieve(a__nats(s(s(0)))) |
| mark(filter(X1, X2, X3)) | → | a__filter(mark(X1), mark(X2), mark(X3)) | | mark(sieve(X)) | → | a__sieve(mark(X)) |
| mark(nats(X)) | → | a__nats(mark(X)) | | mark(zprimes) | → | a__zprimes |
| mark(cons(X1, X2)) | → | cons(mark(X1), X2) | | mark(0) | → | 0 |
| mark(s(X)) | → | s(mark(X)) | | a__filter(X1, X2, X3) | → | filter(X1, X2, X3) |
| a__sieve(X) | → | sieve(X) | | a__nats(X) | → | nats(X) |
| a__zprimes | → | zprimes |
Original Signature
Termination of terms over the following signature is verified: a__filter, nats, 0, sieve, s, a__zprimes, mark, zprimes, a__nats, filter, a__sieve, cons
Strategy
The following SCCs where found
| a__sieve#(cons(s(N), Y)) → mark#(N) | mark#(filter(X1, X2, X3)) → mark#(X3) |
| mark#(cons(X1, X2)) → mark#(X1) | a__filter#(cons(X, Y), s(N), M) → mark#(X) |
| a__nats#(N) → mark#(N) | mark#(sieve(X)) → a__sieve#(mark(X)) |
| mark#(nats(X)) → a__nats#(mark(X)) | mark#(filter(X1, X2, X3)) → mark#(X1) |
| mark#(filter(X1, X2, X3)) → mark#(X2) | mark#(nats(X)) → mark#(X) |
| mark#(s(X)) → mark#(X) | mark#(filter(X1, X2, X3)) → a__filter#(mark(X1), mark(X2), mark(X3)) |
| mark#(sieve(X)) → mark#(X) |
Problem 3: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| mark#(filter(X1, X2, X3)) | → | mark#(X3) | | a__sieve#(cons(s(N), Y)) | → | mark#(N) |
| mark#(cons(X1, X2)) | → | mark#(X1) | | a__filter#(cons(X, Y), s(N), M) | → | mark#(X) |
| a__nats#(N) | → | mark#(N) | | mark#(sieve(X)) | → | a__sieve#(mark(X)) |
| mark#(nats(X)) | → | a__nats#(mark(X)) | | mark#(nats(X)) | → | mark#(X) |
| mark#(filter(X1, X2, X3)) | → | mark#(X2) | | mark#(filter(X1, X2, X3)) | → | mark#(X1) |
| mark#(s(X)) | → | mark#(X) | | mark#(filter(X1, X2, X3)) | → | a__filter#(mark(X1), mark(X2), mark(X3)) |
| mark#(sieve(X)) | → | mark#(X) |
Rewrite Rules
| a__filter(cons(X, Y), 0, M) | → | cons(0, filter(Y, M, M)) | | a__filter(cons(X, Y), s(N), M) | → | cons(mark(X), filter(Y, N, M)) |
| a__sieve(cons(0, Y)) | → | cons(0, sieve(Y)) | | a__sieve(cons(s(N), Y)) | → | cons(s(mark(N)), sieve(filter(Y, N, N))) |
| a__nats(N) | → | cons(mark(N), nats(s(N))) | | a__zprimes | → | a__sieve(a__nats(s(s(0)))) |
| mark(filter(X1, X2, X3)) | → | a__filter(mark(X1), mark(X2), mark(X3)) | | mark(sieve(X)) | → | a__sieve(mark(X)) |
| mark(nats(X)) | → | a__nats(mark(X)) | | mark(zprimes) | → | a__zprimes |
| mark(cons(X1, X2)) | → | cons(mark(X1), X2) | | mark(0) | → | 0 |
| mark(s(X)) | → | s(mark(X)) | | a__filter(X1, X2, X3) | → | filter(X1, X2, X3) |
| a__sieve(X) | → | sieve(X) | | a__nats(X) | → | nats(X) |
| a__zprimes | → | zprimes |
Original Signature
Termination of terms over the following signature is verified: a__filter, nats, 0, sieve, s, a__zprimes, mark, zprimes, a__nats, filter, a__sieve, cons
Strategy
Polynomial Interpretation
- 0: 0
- a__filter(x,y,z): z + y + 2x
- a__filter#(x,y,z): 2x
- a__nats(x): 3x + 1
- a__nats#(x): 3x
- a__sieve(x): x
- a__sieve#(x): 2x
- a__zprimes: 1
- cons(x,y): 2x
- filter(x,y,z): z + y + 2x
- mark(x): x
- mark#(x): 2x
- nats(x): 3x + 1
- s(x): x
- sieve(x): x
- zprimes: 1
Improved Usable rules
| mark(cons(X1, X2)) | → | cons(mark(X1), X2) | | a__nats(N) | → | cons(mark(N), nats(s(N))) |
| mark(0) | → | 0 | | a__nats(X) | → | nats(X) |
| a__sieve(cons(0, Y)) | → | cons(0, sieve(Y)) | | a__zprimes | → | zprimes |
| mark(filter(X1, X2, X3)) | → | a__filter(mark(X1), mark(X2), mark(X3)) | | a__zprimes | → | a__sieve(a__nats(s(s(0)))) |
| a__filter(cons(X, Y), 0, M) | → | cons(0, filter(Y, M, M)) | | a__sieve(cons(s(N), Y)) | → | cons(s(mark(N)), sieve(filter(Y, N, N))) |
| a__filter(cons(X, Y), s(N), M) | → | cons(mark(X), filter(Y, N, M)) | | mark(s(X)) | → | s(mark(X)) |
| mark(sieve(X)) | → | a__sieve(mark(X)) | | mark(zprimes) | → | a__zprimes |
| mark(nats(X)) | → | a__nats(mark(X)) | | a__filter(X1, X2, X3) | → | filter(X1, X2, X3) |
| a__sieve(X) | → | sieve(X) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| mark#(nats(X)) | → | a__nats#(mark(X)) | | mark#(nats(X)) | → | mark#(X) |
Problem 4: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| a__sieve#(cons(s(N), Y)) | → | mark#(N) | | mark#(filter(X1, X2, X3)) | → | mark#(X3) |
| mark#(sieve(X)) | → | a__sieve#(mark(X)) | | mark#(filter(X1, X2, X3)) | → | mark#(X1) |
| mark#(filter(X1, X2, X3)) | → | mark#(X2) | | mark#(cons(X1, X2)) | → | mark#(X1) |
| mark#(s(X)) | → | mark#(X) | | mark#(filter(X1, X2, X3)) | → | a__filter#(mark(X1), mark(X2), mark(X3)) |
| mark#(sieve(X)) | → | mark#(X) | | a__filter#(cons(X, Y), s(N), M) | → | mark#(X) |
| a__nats#(N) | → | mark#(N) |
Rewrite Rules
| a__filter(cons(X, Y), 0, M) | → | cons(0, filter(Y, M, M)) | | a__filter(cons(X, Y), s(N), M) | → | cons(mark(X), filter(Y, N, M)) |
| a__sieve(cons(0, Y)) | → | cons(0, sieve(Y)) | | a__sieve(cons(s(N), Y)) | → | cons(s(mark(N)), sieve(filter(Y, N, N))) |
| a__nats(N) | → | cons(mark(N), nats(s(N))) | | a__zprimes | → | a__sieve(a__nats(s(s(0)))) |
| mark(filter(X1, X2, X3)) | → | a__filter(mark(X1), mark(X2), mark(X3)) | | mark(sieve(X)) | → | a__sieve(mark(X)) |
| mark(nats(X)) | → | a__nats(mark(X)) | | mark(zprimes) | → | a__zprimes |
| mark(cons(X1, X2)) | → | cons(mark(X1), X2) | | mark(0) | → | 0 |
| mark(s(X)) | → | s(mark(X)) | | a__filter(X1, X2, X3) | → | filter(X1, X2, X3) |
| a__sieve(X) | → | sieve(X) | | a__nats(X) | → | nats(X) |
| a__zprimes | → | zprimes |
Original Signature
Termination of terms over the following signature is verified: a__filter, nats, 0, s, sieve, mark, a__zprimes, zprimes, filter, a__nats, cons, a__sieve
Strategy
The following SCCs where found
| a__sieve#(cons(s(N), Y)) → mark#(N) | mark#(filter(X1, X2, X3)) → mark#(X3) |
| mark#(sieve(X)) → a__sieve#(mark(X)) | mark#(cons(X1, X2)) → mark#(X1) |
| mark#(filter(X1, X2, X3)) → mark#(X1) | mark#(filter(X1, X2, X3)) → mark#(X2) |
| mark#(s(X)) → mark#(X) | mark#(filter(X1, X2, X3)) → a__filter#(mark(X1), mark(X2), mark(X3)) |
| a__filter#(cons(X, Y), s(N), M) → mark#(X) | mark#(sieve(X)) → mark#(X) |
Problem 5: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| a__sieve#(cons(s(N), Y)) | → | mark#(N) | | mark#(filter(X1, X2, X3)) | → | mark#(X3) |
| mark#(sieve(X)) | → | a__sieve#(mark(X)) | | mark#(cons(X1, X2)) | → | mark#(X1) |
| mark#(filter(X1, X2, X3)) | → | mark#(X1) | | mark#(filter(X1, X2, X3)) | → | mark#(X2) |
| mark#(s(X)) | → | mark#(X) | | mark#(filter(X1, X2, X3)) | → | a__filter#(mark(X1), mark(X2), mark(X3)) |
| a__filter#(cons(X, Y), s(N), M) | → | mark#(X) | | mark#(sieve(X)) | → | mark#(X) |
Rewrite Rules
| a__filter(cons(X, Y), 0, M) | → | cons(0, filter(Y, M, M)) | | a__filter(cons(X, Y), s(N), M) | → | cons(mark(X), filter(Y, N, M)) |
| a__sieve(cons(0, Y)) | → | cons(0, sieve(Y)) | | a__sieve(cons(s(N), Y)) | → | cons(s(mark(N)), sieve(filter(Y, N, N))) |
| a__nats(N) | → | cons(mark(N), nats(s(N))) | | a__zprimes | → | a__sieve(a__nats(s(s(0)))) |
| mark(filter(X1, X2, X3)) | → | a__filter(mark(X1), mark(X2), mark(X3)) | | mark(sieve(X)) | → | a__sieve(mark(X)) |
| mark(nats(X)) | → | a__nats(mark(X)) | | mark(zprimes) | → | a__zprimes |
| mark(cons(X1, X2)) | → | cons(mark(X1), X2) | | mark(0) | → | 0 |
| mark(s(X)) | → | s(mark(X)) | | a__filter(X1, X2, X3) | → | filter(X1, X2, X3) |
| a__sieve(X) | → | sieve(X) | | a__nats(X) | → | nats(X) |
| a__zprimes | → | zprimes |
Original Signature
Termination of terms over the following signature is verified: a__filter, nats, 0, s, sieve, mark, a__zprimes, zprimes, filter, a__nats, cons, a__sieve
Strategy
Polynomial Interpretation
- 0: 0
- a__filter(x,y,z): z + y + 2x
- a__filter#(x,y,z): 2x
- a__nats(x): 2x
- a__sieve(x): 2x + 2
- a__sieve#(x): x + 2
- a__zprimes: 2
- cons(x,y): x
- filter(x,y,z): z + y + 2x
- mark(x): 2x
- mark#(x): 2x
- nats(x): 2x
- s(x): 3x
- sieve(x): 2x + 1
- zprimes: 1
Improved Usable rules
| mark(cons(X1, X2)) | → | cons(mark(X1), X2) | | a__nats(N) | → | cons(mark(N), nats(s(N))) |
| mark(0) | → | 0 | | a__nats(X) | → | nats(X) |
| a__sieve(cons(0, Y)) | → | cons(0, sieve(Y)) | | a__zprimes | → | zprimes |
| mark(filter(X1, X2, X3)) | → | a__filter(mark(X1), mark(X2), mark(X3)) | | a__zprimes | → | a__sieve(a__nats(s(s(0)))) |
| a__filter(cons(X, Y), 0, M) | → | cons(0, filter(Y, M, M)) | | a__sieve(cons(s(N), Y)) | → | cons(s(mark(N)), sieve(filter(Y, N, N))) |
| a__filter(cons(X, Y), s(N), M) | → | cons(mark(X), filter(Y, N, M)) | | mark(s(X)) | → | s(mark(X)) |
| mark(sieve(X)) | → | a__sieve(mark(X)) | | mark(zprimes) | → | a__zprimes |
| mark(nats(X)) | → | a__nats(mark(X)) | | a__filter(X1, X2, X3) | → | filter(X1, X2, X3) |
| a__sieve(X) | → | sieve(X) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| a__sieve#(cons(s(N), Y)) | → | mark#(N) | | mark#(sieve(X)) | → | mark#(X) |
Problem 6: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| mark#(filter(X1, X2, X3)) | → | mark#(X3) | | mark#(sieve(X)) | → | a__sieve#(mark(X)) |
| mark#(filter(X1, X2, X3)) | → | mark#(X2) | | mark#(filter(X1, X2, X3)) | → | mark#(X1) |
| mark#(cons(X1, X2)) | → | mark#(X1) | | mark#(s(X)) | → | mark#(X) |
| mark#(filter(X1, X2, X3)) | → | a__filter#(mark(X1), mark(X2), mark(X3)) | | a__filter#(cons(X, Y), s(N), M) | → | mark#(X) |
Rewrite Rules
| a__filter(cons(X, Y), 0, M) | → | cons(0, filter(Y, M, M)) | | a__filter(cons(X, Y), s(N), M) | → | cons(mark(X), filter(Y, N, M)) |
| a__sieve(cons(0, Y)) | → | cons(0, sieve(Y)) | | a__sieve(cons(s(N), Y)) | → | cons(s(mark(N)), sieve(filter(Y, N, N))) |
| a__nats(N) | → | cons(mark(N), nats(s(N))) | | a__zprimes | → | a__sieve(a__nats(s(s(0)))) |
| mark(filter(X1, X2, X3)) | → | a__filter(mark(X1), mark(X2), mark(X3)) | | mark(sieve(X)) | → | a__sieve(mark(X)) |
| mark(nats(X)) | → | a__nats(mark(X)) | | mark(zprimes) | → | a__zprimes |
| mark(cons(X1, X2)) | → | cons(mark(X1), X2) | | mark(0) | → | 0 |
| mark(s(X)) | → | s(mark(X)) | | a__filter(X1, X2, X3) | → | filter(X1, X2, X3) |
| a__sieve(X) | → | sieve(X) | | a__nats(X) | → | nats(X) |
| a__zprimes | → | zprimes |
Original Signature
Termination of terms over the following signature is verified: a__filter, nats, 0, sieve, s, a__zprimes, mark, zprimes, a__nats, filter, a__sieve, cons
Strategy
The following SCCs where found
| mark#(filter(X1, X2, X3)) → mark#(X3) | mark#(cons(X1, X2)) → mark#(X1) |
| mark#(filter(X1, X2, X3)) → mark#(X1) | mark#(filter(X1, X2, X3)) → mark#(X2) |
| mark#(s(X)) → mark#(X) | mark#(filter(X1, X2, X3)) → a__filter#(mark(X1), mark(X2), mark(X3)) |
| a__filter#(cons(X, Y), s(N), M) → mark#(X) |
Problem 7: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| mark#(filter(X1, X2, X3)) | → | mark#(X3) | | mark#(cons(X1, X2)) | → | mark#(X1) |
| mark#(filter(X1, X2, X3)) | → | mark#(X1) | | mark#(filter(X1, X2, X3)) | → | mark#(X2) |
| mark#(s(X)) | → | mark#(X) | | mark#(filter(X1, X2, X3)) | → | a__filter#(mark(X1), mark(X2), mark(X3)) |
| a__filter#(cons(X, Y), s(N), M) | → | mark#(X) |
Rewrite Rules
| a__filter(cons(X, Y), 0, M) | → | cons(0, filter(Y, M, M)) | | a__filter(cons(X, Y), s(N), M) | → | cons(mark(X), filter(Y, N, M)) |
| a__sieve(cons(0, Y)) | → | cons(0, sieve(Y)) | | a__sieve(cons(s(N), Y)) | → | cons(s(mark(N)), sieve(filter(Y, N, N))) |
| a__nats(N) | → | cons(mark(N), nats(s(N))) | | a__zprimes | → | a__sieve(a__nats(s(s(0)))) |
| mark(filter(X1, X2, X3)) | → | a__filter(mark(X1), mark(X2), mark(X3)) | | mark(sieve(X)) | → | a__sieve(mark(X)) |
| mark(nats(X)) | → | a__nats(mark(X)) | | mark(zprimes) | → | a__zprimes |
| mark(cons(X1, X2)) | → | cons(mark(X1), X2) | | mark(0) | → | 0 |
| mark(s(X)) | → | s(mark(X)) | | a__filter(X1, X2, X3) | → | filter(X1, X2, X3) |
| a__sieve(X) | → | sieve(X) | | a__nats(X) | → | nats(X) |
| a__zprimes | → | zprimes |
Original Signature
Termination of terms over the following signature is verified: a__filter, nats, 0, sieve, s, a__zprimes, mark, zprimes, a__nats, filter, a__sieve, cons
Strategy
Polynomial Interpretation
- 0: 0
- a__filter(x,y,z): z + y + 2x + 1
- a__filter#(x,y,z): 2x
- a__nats(x): 2x + 1
- a__sieve(x): 2x
- a__zprimes: 2
- cons(x,y): x + 1
- filter(x,y,z): z + y + 2x + 1
- mark(x): 2x
- mark#(x): 2x
- nats(x): 2x + 1
- s(x): 2x
- sieve(x): 2x
- zprimes: 1
Improved Usable rules
| mark(cons(X1, X2)) | → | cons(mark(X1), X2) | | a__nats(N) | → | cons(mark(N), nats(s(N))) |
| mark(0) | → | 0 | | a__nats(X) | → | nats(X) |
| a__sieve(cons(0, Y)) | → | cons(0, sieve(Y)) | | a__zprimes | → | zprimes |
| mark(filter(X1, X2, X3)) | → | a__filter(mark(X1), mark(X2), mark(X3)) | | a__zprimes | → | a__sieve(a__nats(s(s(0)))) |
| a__filter(cons(X, Y), 0, M) | → | cons(0, filter(Y, M, M)) | | a__sieve(cons(s(N), Y)) | → | cons(s(mark(N)), sieve(filter(Y, N, N))) |
| a__filter(cons(X, Y), s(N), M) | → | cons(mark(X), filter(Y, N, M)) | | mark(s(X)) | → | s(mark(X)) |
| mark(sieve(X)) | → | a__sieve(mark(X)) | | mark(zprimes) | → | a__zprimes |
| mark(nats(X)) | → | a__nats(mark(X)) | | a__filter(X1, X2, X3) | → | filter(X1, X2, X3) |
| a__sieve(X) | → | sieve(X) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| mark#(filter(X1, X2, X3)) | → | mark#(X3) | | mark#(filter(X1, X2, X3)) | → | mark#(X2) |
| mark#(filter(X1, X2, X3)) | → | mark#(X1) | | mark#(cons(X1, X2)) | → | mark#(X1) |
| mark#(filter(X1, X2, X3)) | → | a__filter#(mark(X1), mark(X2), mark(X3)) | | a__filter#(cons(X, Y), s(N), M) | → | mark#(X) |
Problem 8: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
Rewrite Rules
| a__filter(cons(X, Y), 0, M) | → | cons(0, filter(Y, M, M)) | | a__filter(cons(X, Y), s(N), M) | → | cons(mark(X), filter(Y, N, M)) |
| a__sieve(cons(0, Y)) | → | cons(0, sieve(Y)) | | a__sieve(cons(s(N), Y)) | → | cons(s(mark(N)), sieve(filter(Y, N, N))) |
| a__nats(N) | → | cons(mark(N), nats(s(N))) | | a__zprimes | → | a__sieve(a__nats(s(s(0)))) |
| mark(filter(X1, X2, X3)) | → | a__filter(mark(X1), mark(X2), mark(X3)) | | mark(sieve(X)) | → | a__sieve(mark(X)) |
| mark(nats(X)) | → | a__nats(mark(X)) | | mark(zprimes) | → | a__zprimes |
| mark(cons(X1, X2)) | → | cons(mark(X1), X2) | | mark(0) | → | 0 |
| mark(s(X)) | → | s(mark(X)) | | a__filter(X1, X2, X3) | → | filter(X1, X2, X3) |
| a__sieve(X) | → | sieve(X) | | a__nats(X) | → | nats(X) |
| a__zprimes | → | zprimes |
Original Signature
Termination of terms over the following signature is verified: a__filter, nats, 0, s, sieve, mark, a__zprimes, zprimes, filter, a__nats, cons, a__sieve
Strategy
Polynomial Interpretation
- 0: 0
- a__filter(x,y,z): 0
- a__nats(x): 0
- a__sieve(x): 0
- a__zprimes: 0
- cons(x,y): 0
- filter(x,y,z): 0
- mark(x): 0
- mark#(x): x
- nats(x): 0
- s(x): x + 1
- sieve(x): 0
- zprimes: 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: