The rewrite relation of the following TRS is considered.
a__filter(cons(X,Y),0,M) | → | cons(0,filter(Y,M,M)) | (1) |
a__filter(cons(X,Y),s(N),M) | → | cons(mark(X),filter(Y,N,M)) | (2) |
a__sieve(cons(0,Y)) | → | cons(0,sieve(Y)) | (3) |
a__sieve(cons(s(N),Y)) | → | cons(s(mark(N)),sieve(filter(Y,N,N))) | (4) |
a__nats(N) | → | cons(mark(N),nats(s(N))) | (5) |
a__zprimes | → | a__sieve(a__nats(s(s(0)))) | (6) |
mark(filter(X1,X2,X3)) | → | a__filter(mark(X1),mark(X2),mark(X3)) | (7) |
mark(sieve(X)) | → | a__sieve(mark(X)) | (8) |
mark(nats(X)) | → | a__nats(mark(X)) | (9) |
mark(zprimes) | → | a__zprimes | (10) |
mark(cons(X1,X2)) | → | cons(mark(X1),X2) | (11) |
mark(0) | → | 0 | (12) |
mark(s(X)) | → | s(mark(X)) | (13) |
a__filter(X1,X2,X3) | → | filter(X1,X2,X3) | (14) |
a__sieve(X) | → | sieve(X) | (15) |
a__nats(X) | → | nats(X) | (16) |
a__zprimes | → | zprimes | (17) |
a__filter#(cons(X,Y),s(N),M) | → | mark#(X) | (18) |
a__sieve#(cons(s(N),Y)) | → | mark#(N) | (19) |
a__nats#(N) | → | mark#(N) | (20) |
a__zprimes# | → | a__nats#(s(s(0))) | (21) |
a__zprimes# | → | a__sieve#(a__nats(s(s(0)))) | (22) |
mark#(filter(X1,X2,X3)) | → | mark#(X3) | (23) |
mark#(filter(X1,X2,X3)) | → | mark#(X2) | (24) |
mark#(filter(X1,X2,X3)) | → | mark#(X1) | (25) |
mark#(filter(X1,X2,X3)) | → | a__filter#(mark(X1),mark(X2),mark(X3)) | (26) |
mark#(sieve(X)) | → | mark#(X) | (27) |
mark#(sieve(X)) | → | a__sieve#(mark(X)) | (28) |
mark#(nats(X)) | → | mark#(X) | (29) |
mark#(nats(X)) | → | a__nats#(mark(X)) | (30) |
mark#(zprimes) | → | a__zprimes# | (31) |
mark#(cons(X1,X2)) | → | mark#(X1) | (32) |
mark#(s(X)) | → | mark#(X) | (33) |
[s(x1)] | = | 0 · x1 + 0 |
[a__nats#(x1)] | = | 0 · x1 + 0 |
[a__nats(x1)] | = | 0 · x1 + 0 |
[0] | = | 0 |
[filter(x1, x2, x3)] | = | 0 · x1 + 0 · x2 + 0 · x3 + 0 |
[a__zprimes#] | = | 0 |
[mark(x1)] | = | 0 · x1 + 0 |
[a__zprimes] | = | 2 |
[cons(x1, x2)] | = | 0 · x1 + -∞ · x2 + 0 |
[sieve(x1)] | = | 0 · x1 + 0 |
[a__sieve#(x1)] | = | 0 · x1 + 0 |
[a__filter#(x1, x2, x3)] | = | 0 · x1 + 0 · x2 + 0 · x3 + 0 |
[zprimes] | = | 2 |
[nats(x1)] | = | 0 · x1 + -∞ |
[mark#(x1)] | = | 0 · x1 + 0 |
[a__sieve(x1)] | = | 0 · x1 + 0 |
[a__filter(x1, x2, x3)] | = | 0 · x1 + 0 · x2 + 0 · x3 + 0 |
a__filter(cons(X,Y),0,M) | → | cons(0,filter(Y,M,M)) | (1) |
a__filter(cons(X,Y),s(N),M) | → | cons(mark(X),filter(Y,N,M)) | (2) |
a__sieve(cons(0,Y)) | → | cons(0,sieve(Y)) | (3) |
a__sieve(cons(s(N),Y)) | → | cons(s(mark(N)),sieve(filter(Y,N,N))) | (4) |
a__nats(N) | → | cons(mark(N),nats(s(N))) | (5) |
a__zprimes | → | a__sieve(a__nats(s(s(0)))) | (6) |
mark(filter(X1,X2,X3)) | → | a__filter(mark(X1),mark(X2),mark(X3)) | (7) |
mark(sieve(X)) | → | a__sieve(mark(X)) | (8) |
mark(nats(X)) | → | a__nats(mark(X)) | (9) |
mark(zprimes) | → | a__zprimes | (10) |
mark(cons(X1,X2)) | → | cons(mark(X1),X2) | (11) |
mark(0) | → | 0 | (12) |
mark(s(X)) | → | s(mark(X)) | (13) |
a__filter(X1,X2,X3) | → | filter(X1,X2,X3) | (14) |
a__sieve(X) | → | sieve(X) | (15) |
a__nats(X) | → | nats(X) | (16) |
a__zprimes | → | zprimes | (17) |
mark#(zprimes) | → | a__zprimes# | (31) |
The dependency pairs are split into 1 component.
a__nats#(N) | → | mark#(N) | (20) |
mark#(filter(X1,X2,X3)) | → | mark#(X3) | (23) |
mark#(filter(X1,X2,X3)) | → | mark#(X2) | (24) |
mark#(filter(X1,X2,X3)) | → | mark#(X1) | (25) |
mark#(filter(X1,X2,X3)) | → | a__filter#(mark(X1),mark(X2),mark(X3)) | (26) |
a__filter#(cons(X,Y),s(N),M) | → | mark#(X) | (18) |
mark#(sieve(X)) | → | mark#(X) | (27) |
mark#(sieve(X)) | → | a__sieve#(mark(X)) | (28) |
a__sieve#(cons(s(N),Y)) | → | mark#(N) | (19) |
mark#(nats(X)) | → | mark#(X) | (29) |
mark#(nats(X)) | → | a__nats#(mark(X)) | (30) |
mark#(cons(X1,X2)) | → | mark#(X1) | (32) |
mark#(s(X)) | → | mark#(X) | (33) |
[s(x1)] | = | 0 · x1 + 0 |
[a__nats#(x1)] | = | 2 · x1 + 2 |
[a__nats(x1)] | = | 0 · x1 + 0 |
[0] | = | 0 |
[filter(x1, x2, x3)] | = | 0 · x1 + 0 · x2 + 0 · x3 + 0 |
[mark(x1)] | = | 0 · x1 + 0 |
[a__zprimes] | = | 4 |
[cons(x1, x2)] | = | 0 · x1 + 0 · x2 + -∞ |
[sieve(x1)] | = | 2 · x1 + 2 |
[a__sieve#(x1)] | = | 3 · x1 + 0 |
[a__filter#(x1, x2, x3)] | = | 2 · x1 + 2 · x2 + 2 · x3 + 0 |
[zprimes] | = | 4 |
[nats(x1)] | = | 0 · x1 + -∞ |
[mark#(x1)] | = | 2 · x1 + 2 |
[a__sieve(x1)] | = | 2 · x1 + 2 |
[a__filter(x1, x2, x3)] | = | 0 · x1 + 0 · x2 + 0 · x3 + 0 |
a__filter(cons(X,Y),0,M) | → | cons(0,filter(Y,M,M)) | (1) |
a__filter(cons(X,Y),s(N),M) | → | cons(mark(X),filter(Y,N,M)) | (2) |
a__sieve(cons(0,Y)) | → | cons(0,sieve(Y)) | (3) |
a__sieve(cons(s(N),Y)) | → | cons(s(mark(N)),sieve(filter(Y,N,N))) | (4) |
a__nats(N) | → | cons(mark(N),nats(s(N))) | (5) |
a__zprimes | → | a__sieve(a__nats(s(s(0)))) | (6) |
mark(filter(X1,X2,X3)) | → | a__filter(mark(X1),mark(X2),mark(X3)) | (7) |
mark(sieve(X)) | → | a__sieve(mark(X)) | (8) |
mark(nats(X)) | → | a__nats(mark(X)) | (9) |
mark(zprimes) | → | a__zprimes | (10) |
mark(cons(X1,X2)) | → | cons(mark(X1),X2) | (11) |
mark(0) | → | 0 | (12) |
mark(s(X)) | → | s(mark(X)) | (13) |
a__filter(X1,X2,X3) | → | filter(X1,X2,X3) | (14) |
a__sieve(X) | → | sieve(X) | (15) |
a__nats(X) | → | nats(X) | (16) |
a__zprimes | → | zprimes | (17) |
mark#(sieve(X)) | → | mark#(X) | (27) |
mark#(sieve(X)) | → | a__sieve#(mark(X)) | (28) |
a__sieve#(cons(s(N),Y)) | → | mark#(N) | (19) |
[s(x1)] | = | 0 · x1 + 0 |
[a__nats#(x1)] | = | 0 · x1 + 0 |
[a__nats(x1)] | = | 1 · x1 + 1 |
[0] | = | 0 |
[filter(x1, x2, x3)] | = | 0 · x1 + 0 · x2 + 0 · x3 + 0 |
[mark(x1)] | = | 0 · x1 + 0 |
[a__zprimes] | = | 1 |
[cons(x1, x2)] | = | 0 · x1 + 0 · x2 + -∞ |
[sieve(x1)] | = | 0 · x1 + 0 |
[a__filter#(x1, x2, x3)] | = | 0 · x1 + 0 · x2 + 0 · x3 + -∞ |
[zprimes] | = | 1 |
[nats(x1)] | = | 1 · x1 + 1 |
[mark#(x1)] | = | 0 · x1 + 0 |
[a__sieve(x1)] | = | 0 · x1 + 0 |
[a__filter(x1, x2, x3)] | = | 0 · x1 + 0 · x2 + 0 · x3 + 0 |
a__filter(cons(X,Y),0,M) | → | cons(0,filter(Y,M,M)) | (1) |
a__filter(cons(X,Y),s(N),M) | → | cons(mark(X),filter(Y,N,M)) | (2) |
a__sieve(cons(0,Y)) | → | cons(0,sieve(Y)) | (3) |
a__sieve(cons(s(N),Y)) | → | cons(s(mark(N)),sieve(filter(Y,N,N))) | (4) |
a__nats(N) | → | cons(mark(N),nats(s(N))) | (5) |
a__zprimes | → | a__sieve(a__nats(s(s(0)))) | (6) |
mark(filter(X1,X2,X3)) | → | a__filter(mark(X1),mark(X2),mark(X3)) | (7) |
mark(sieve(X)) | → | a__sieve(mark(X)) | (8) |
mark(nats(X)) | → | a__nats(mark(X)) | (9) |
mark(zprimes) | → | a__zprimes | (10) |
mark(cons(X1,X2)) | → | cons(mark(X1),X2) | (11) |
mark(0) | → | 0 | (12) |
mark(s(X)) | → | s(mark(X)) | (13) |
a__filter(X1,X2,X3) | → | filter(X1,X2,X3) | (14) |
a__sieve(X) | → | sieve(X) | (15) |
a__nats(X) | → | nats(X) | (16) |
a__zprimes | → | zprimes | (17) |
mark#(nats(X)) | → | mark#(X) | (29) |
mark#(nats(X)) | → | a__nats#(mark(X)) | (30) |
The dependency pairs are split into 1 component.
mark#(filter(X1,X2,X3)) | → | mark#(X3) | (23) |
mark#(filter(X1,X2,X3)) | → | mark#(X2) | (24) |
mark#(filter(X1,X2,X3)) | → | mark#(X1) | (25) |
mark#(filter(X1,X2,X3)) | → | a__filter#(mark(X1),mark(X2),mark(X3)) | (26) |
a__filter#(cons(X,Y),s(N),M) | → | mark#(X) | (18) |
mark#(cons(X1,X2)) | → | mark#(X1) | (32) |
mark#(s(X)) | → | mark#(X) | (33) |
[s(x1)] | = | 0 · x1 + 0 |
[a__nats(x1)] | = | 5 · x1 + 0 |
[0] | = | 2 |
[filter(x1, x2, x3)] | = | 2 · x1 + 0 · x2 + 0 · x3 + -∞ |
[mark(x1)] | = | 0 · x1 + -∞ |
[a__zprimes] | = | 7 |
[cons(x1, x2)] | = | 0 · x1 + -∞ · x2 + 0 |
[sieve(x1)] | = | 0 · x1 + 6 |
[a__filter#(x1, x2, x3)] | = | 0 · x1 + -∞ · x2 + -∞ · x3 + -∞ |
[zprimes] | = | 7 |
[nats(x1)] | = | 5 · x1 + 0 |
[mark#(x1)] | = | 0 · x1 + 0 |
[a__sieve(x1)] | = | 0 · x1 + 6 |
[a__filter(x1, x2, x3)] | = | 2 · x1 + 0 · x2 + 0 · x3 + -∞ |
a__filter(cons(X,Y),0,M) | → | cons(0,filter(Y,M,M)) | (1) |
a__filter(cons(X,Y),s(N),M) | → | cons(mark(X),filter(Y,N,M)) | (2) |
a__sieve(cons(0,Y)) | → | cons(0,sieve(Y)) | (3) |
a__sieve(cons(s(N),Y)) | → | cons(s(mark(N)),sieve(filter(Y,N,N))) | (4) |
a__nats(N) | → | cons(mark(N),nats(s(N))) | (5) |
a__zprimes | → | a__sieve(a__nats(s(s(0)))) | (6) |
mark(filter(X1,X2,X3)) | → | a__filter(mark(X1),mark(X2),mark(X3)) | (7) |
mark(sieve(X)) | → | a__sieve(mark(X)) | (8) |
mark(nats(X)) | → | a__nats(mark(X)) | (9) |
mark(zprimes) | → | a__zprimes | (10) |
mark(cons(X1,X2)) | → | cons(mark(X1),X2) | (11) |
mark(0) | → | 0 | (12) |
mark(s(X)) | → | s(mark(X)) | (13) |
a__filter(X1,X2,X3) | → | filter(X1,X2,X3) | (14) |
a__sieve(X) | → | sieve(X) | (15) |
a__nats(X) | → | nats(X) | (16) |
a__zprimes | → | zprimes | (17) |
mark#(filter(X1,X2,X3)) | → | a__filter#(mark(X1),mark(X2),mark(X3)) | (26) |
The dependency pairs are split into 1 component.
mark#(s(X)) | → | mark#(X) | (33) |
mark#(filter(X1,X2,X3)) | → | mark#(X3) | (23) |
mark#(filter(X1,X2,X3)) | → | mark#(X2) | (24) |
mark#(filter(X1,X2,X3)) | → | mark#(X1) | (25) |
mark#(cons(X1,X2)) | → | mark#(X1) | (32) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
mark#(s(X)) | → | mark#(X) | (33) |
1 | > | 1 | |
mark#(filter(X1,X2,X3)) | → | mark#(X3) | (23) |
1 | > | 1 | |
mark#(filter(X1,X2,X3)) | → | mark#(X2) | (24) |
1 | > | 1 | |
mark#(filter(X1,X2,X3)) | → | mark#(X1) | (25) |
1 | > | 1 | |
mark#(cons(X1,X2)) | → | mark#(X1) | (32) |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.