Term Rewriting System R:
[X, Y, M, N, X1, X2, X3]
afilter(cons(X, Y), 0, M) -> cons(0, filter(Y, M, M))
afilter(cons(X, Y), s(N), M) -> cons(mark(X), filter(Y, N, M))
afilter(X1, X2, X3) -> filter(X1, X2, X3)
asieve(cons(0, Y)) -> cons(0, sieve(Y))
asieve(cons(s(N), Y)) -> cons(s(mark(N)), sieve(filter(Y, N, N)))
asieve(X) -> sieve(X)
anats(N) -> cons(mark(N), nats(s(N)))
anats(X) -> nats(X)
azprimes -> asieve(anats(s(s(0))))
azprimes -> zprimes
mark(filter(X1, X2, X3)) -> afilter(mark(X1), mark(X2), mark(X3))
mark(sieve(X)) -> asieve(mark(X))
mark(nats(X)) -> anats(mark(X))
mark(zprimes) -> azprimes
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0) -> 0
mark(s(X)) -> s(mark(X))

Innermost Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

AFILTER(cons(X, Y), s(N), M) -> MARK(X)
ASIEVE(cons(s(N), Y)) -> MARK(N)
ANATS(N) -> MARK(N)
AZPRIMES -> ASIEVE(anats(s(s(0))))
AZPRIMES -> ANATS(s(s(0)))
MARK(filter(X1, X2, X3)) -> AFILTER(mark(X1), mark(X2), mark(X3))
MARK(filter(X1, X2, X3)) -> MARK(X1)
MARK(filter(X1, X2, X3)) -> MARK(X2)
MARK(filter(X1, X2, X3)) -> MARK(X3)
MARK(sieve(X)) -> ASIEVE(mark(X))
MARK(sieve(X)) -> MARK(X)
MARK(nats(X)) -> ANATS(mark(X))
MARK(nats(X)) -> MARK(X)
MARK(zprimes) -> AZPRIMES
MARK(cons(X1, X2)) -> MARK(X1)
MARK(s(X)) -> MARK(X)

Furthermore, R contains one SCC.


   R
DPs
       →DP Problem 1
Negative Polynomial Order


Dependency Pairs:

MARK(s(X)) -> MARK(X)
MARK(cons(X1, X2)) -> MARK(X1)
AZPRIMES -> ANATS(s(s(0)))
AZPRIMES -> ASIEVE(anats(s(s(0))))
MARK(zprimes) -> AZPRIMES
MARK(nats(X)) -> MARK(X)
ANATS(N) -> MARK(N)
MARK(nats(X)) -> ANATS(mark(X))
MARK(sieve(X)) -> MARK(X)
ASIEVE(cons(s(N), Y)) -> MARK(N)
MARK(sieve(X)) -> ASIEVE(mark(X))
MARK(filter(X1, X2, X3)) -> MARK(X3)
MARK(filter(X1, X2, X3)) -> MARK(X2)
MARK(filter(X1, X2, X3)) -> MARK(X1)
MARK(filter(X1, X2, X3)) -> AFILTER(mark(X1), mark(X2), mark(X3))
AFILTER(cons(X, Y), s(N), M) -> MARK(X)


Rules:


afilter(cons(X, Y), 0, M) -> cons(0, filter(Y, M, M))
afilter(cons(X, Y), s(N), M) -> cons(mark(X), filter(Y, N, M))
afilter(X1, X2, X3) -> filter(X1, X2, X3)
asieve(cons(0, Y)) -> cons(0, sieve(Y))
asieve(cons(s(N), Y)) -> cons(s(mark(N)), sieve(filter(Y, N, N)))
asieve(X) -> sieve(X)
anats(N) -> cons(mark(N), nats(s(N)))
anats(X) -> nats(X)
azprimes -> asieve(anats(s(s(0))))
azprimes -> zprimes
mark(filter(X1, X2, X3)) -> afilter(mark(X1), mark(X2), mark(X3))
mark(sieve(X)) -> asieve(mark(X))
mark(nats(X)) -> anats(mark(X))
mark(zprimes) -> azprimes
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0) -> 0
mark(s(X)) -> s(mark(X))


Strategy:

innermost




The following Dependency Pairs can be strictly oriented using the given order.

MARK(cons(X1, X2)) -> MARK(X1)
AZPRIMES -> ANATS(s(s(0)))
MARK(nats(X)) -> MARK(X)
MARK(nats(X)) -> ANATS(mark(X))
ASIEVE(cons(s(N), Y)) -> MARK(N)
AFILTER(cons(X, Y), s(N), M) -> MARK(X)


Moreover, the following usable rules (regarding the implicit AFS) are oriented.

asieve(cons(s(N), Y)) -> cons(s(mark(N)), sieve(filter(Y, N, N)))
asieve(X) -> sieve(X)
mark(s(X)) -> s(mark(X))
afilter(X1, X2, X3) -> filter(X1, X2, X3)
mark(filter(X1, X2, X3)) -> afilter(mark(X1), mark(X2), mark(X3))
afilter(cons(X, Y), 0, M) -> cons(0, filter(Y, M, M))
anats(X) -> nats(X)
azprimes -> zprimes
anats(N) -> cons(mark(N), nats(s(N)))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(zprimes) -> azprimes
mark(nats(X)) -> anats(mark(X))
afilter(cons(X, Y), s(N), M) -> cons(mark(X), filter(Y, N, M))
azprimes -> asieve(anats(s(s(0))))
mark(sieve(X)) -> asieve(mark(X))
mark(0) -> 0
asieve(cons(0, Y)) -> cons(0, sieve(Y))


Used ordering:
Polynomial Order with Interpretation:

POL( MARK(x1) ) = x1

POL( cons(x1, x2) ) = x1 + 1

POL( filter(x1, ..., x3) ) = x1 + x2 + x3

POL( AFILTER(x1, ..., x3) ) = x1

POL( mark(x1) ) = x1

POL( nats(x1) ) = x1 + 1

POL( ANATS(x1) ) = x1

POL( ASIEVE(x1) ) = x1

POL( s(x1) ) = x1

POL( sieve(x1) ) = x1

POL( AZPRIMES ) = 1

POL( 0 ) = 0

POL( zprimes ) = 1

POL( anats(x1) ) = x1 + 1

POL( asieve(x1) ) = x1

POL( afilter(x1, ..., x3) ) = x1 + x2 + x3

POL( azprimes ) = 1


This results in one new DP problem.


   R
DPs
       →DP Problem 1
Neg POLO
           →DP Problem 2
Dependency Graph


Dependency Pairs:

MARK(s(X)) -> MARK(X)
AZPRIMES -> ASIEVE(anats(s(s(0))))
MARK(zprimes) -> AZPRIMES
ANATS(N) -> MARK(N)
MARK(sieve(X)) -> MARK(X)
MARK(sieve(X)) -> ASIEVE(mark(X))
MARK(filter(X1, X2, X3)) -> MARK(X3)
MARK(filter(X1, X2, X3)) -> MARK(X2)
MARK(filter(X1, X2, X3)) -> MARK(X1)
MARK(filter(X1, X2, X3)) -> AFILTER(mark(X1), mark(X2), mark(X3))


Rules:


afilter(cons(X, Y), 0, M) -> cons(0, filter(Y, M, M))
afilter(cons(X, Y), s(N), M) -> cons(mark(X), filter(Y, N, M))
afilter(X1, X2, X3) -> filter(X1, X2, X3)
asieve(cons(0, Y)) -> cons(0, sieve(Y))
asieve(cons(s(N), Y)) -> cons(s(mark(N)), sieve(filter(Y, N, N)))
asieve(X) -> sieve(X)
anats(N) -> cons(mark(N), nats(s(N)))
anats(X) -> nats(X)
azprimes -> asieve(anats(s(s(0))))
azprimes -> zprimes
mark(filter(X1, X2, X3)) -> afilter(mark(X1), mark(X2), mark(X3))
mark(sieve(X)) -> asieve(mark(X))
mark(nats(X)) -> anats(mark(X))
mark(zprimes) -> azprimes
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0) -> 0
mark(s(X)) -> s(mark(X))


Strategy:

innermost




Using the Dependency Graph the DP problem was split into 1 DP problems.


   R
DPs
       →DP Problem 1
Neg POLO
           →DP Problem 2
DGraph
             ...
               →DP Problem 3
Usable Rules (Innermost)


Dependency Pairs:

MARK(sieve(X)) -> MARK(X)
MARK(filter(X1, X2, X3)) -> MARK(X3)
MARK(filter(X1, X2, X3)) -> MARK(X2)
MARK(filter(X1, X2, X3)) -> MARK(X1)
MARK(s(X)) -> MARK(X)


Rules:


afilter(cons(X, Y), 0, M) -> cons(0, filter(Y, M, M))
afilter(cons(X, Y), s(N), M) -> cons(mark(X), filter(Y, N, M))
afilter(X1, X2, X3) -> filter(X1, X2, X3)
asieve(cons(0, Y)) -> cons(0, sieve(Y))
asieve(cons(s(N), Y)) -> cons(s(mark(N)), sieve(filter(Y, N, N)))
asieve(X) -> sieve(X)
anats(N) -> cons(mark(N), nats(s(N)))
anats(X) -> nats(X)
azprimes -> asieve(anats(s(s(0))))
azprimes -> zprimes
mark(filter(X1, X2, X3)) -> afilter(mark(X1), mark(X2), mark(X3))
mark(sieve(X)) -> asieve(mark(X))
mark(nats(X)) -> anats(mark(X))
mark(zprimes) -> azprimes
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0) -> 0
mark(s(X)) -> s(mark(X))


Strategy:

innermost




As we are in the innermost case, we can delete all 17 non-usable-rules.


   R
DPs
       →DP Problem 1
Neg POLO
           →DP Problem 2
DGraph
             ...
               →DP Problem 4
Size-Change Principle


Dependency Pairs:

MARK(sieve(X)) -> MARK(X)
MARK(filter(X1, X2, X3)) -> MARK(X3)
MARK(filter(X1, X2, X3)) -> MARK(X2)
MARK(filter(X1, X2, X3)) -> MARK(X1)
MARK(s(X)) -> MARK(X)


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. MARK(sieve(X)) -> MARK(X)
  2. MARK(filter(X1, X2, X3)) -> MARK(X3)
  3. MARK(filter(X1, X2, X3)) -> MARK(X2)
  4. MARK(filter(X1, X2, X3)) -> MARK(X1)
  5. MARK(s(X)) -> MARK(X)
and get the following Size-Change Graph(s):
{1, 2, 3, 4, 5} , {1, 2, 3, 4, 5}
1>1

which lead(s) to this/these maximal multigraph(s):
{1, 2, 3, 4, 5} , {1, 2, 3, 4, 5}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
filter(x1, x2, x3) -> filter(x1, x2, x3)
sieve(x1) -> sieve(x1)
s(x1) -> s(x1)

We obtain no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:05 minutes