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__zprimes# | → | a__sieve#(a__nats(s(s(0)))) | (18) |
mark#(nats(X)) | → | mark#(X) | (19) |
a__nats#(N) | → | mark#(N) | (20) |
mark#(filter(X1,X2,X3)) | → | mark#(X1) | (21) |
mark#(filter(X1,X2,X3)) | → | mark#(X3) | (22) |
mark#(zprimes) | → | a__zprimes# | (23) |
mark#(sieve(X)) | → | a__sieve#(mark(X)) | (24) |
mark#(cons(X1,X2)) | → | mark#(X1) | (25) |
mark#(filter(X1,X2,X3)) | → | mark#(X2) | (26) |
mark#(nats(X)) | → | a__nats#(mark(X)) | (27) |
a__sieve#(cons(s(N),Y)) | → | mark#(N) | (28) |
a__zprimes# | → | a__nats#(s(s(0))) | (29) |
a__filter#(cons(X,Y),s(N),M) | → | mark#(X) | (30) |
mark#(filter(X1,X2,X3)) | → | a__filter#(mark(X1),mark(X2),mark(X3)) | (31) |
mark#(sieve(X)) | → | mark#(X) | (32) |
mark#(s(X)) | → | mark#(X) | (33) |
The dependency pairs are split into 1 component.
mark#(s(X)) | → | mark#(X) | (33) |
mark#(sieve(X)) | → | a__sieve#(mark(X)) | (24) |
mark#(sieve(X)) | → | mark#(X) | (32) |
mark#(zprimes) | → | a__zprimes# | (23) |
mark#(filter(X1,X2,X3)) | → | a__filter#(mark(X1),mark(X2),mark(X3)) | (31) |
a__filter#(cons(X,Y),s(N),M) | → | mark#(X) | (30) |
a__zprimes# | → | a__nats#(s(s(0))) | (29) |
a__sieve#(cons(s(N),Y)) | → | mark#(N) | (28) |
mark#(filter(X1,X2,X3)) | → | mark#(X3) | (22) |
mark#(filter(X1,X2,X3)) | → | mark#(X1) | (21) |
a__nats#(N) | → | mark#(N) | (20) |
mark#(nats(X)) | → | mark#(X) | (19) |
mark#(nats(X)) | → | a__nats#(mark(X)) | (27) |
mark#(filter(X1,X2,X3)) | → | mark#(X2) | (26) |
mark#(cons(X1,X2)) | → | mark#(X1) | (25) |
a__zprimes# | → | a__sieve#(a__nats(s(s(0)))) | (18) |
[zprimes] | = | 23681 |
[s(x1)] | = | x1 + 1 |
[a__zprimes] | = | 23681 |
[a__nats(x1)] | = | x1 + 2 |
[a__filter#(x1, x2, x3)] | = | x1 + x3 + 1 |
[a__sieve(x1)] | = | x1 + 1 |
[a__zprimes#] | = | 23681 |
[mark#(x1)] | = | x1 + 1 |
[0] | = | 23676 |
[a__nats#(x1)] | = | x1 + 2 |
[mark(x1)] | = | x1 + 0 |
[sieve(x1)] | = | x1 + 1 |
[nats(x1)] | = | x1 + 2 |
[cons(x1, x2)] | = | x1 + 1 |
[filter(x1, x2, x3)] | = | x1 + x2 + x3 + 4 |
[a__sieve#(x1)] | = | x1 + 0 |
[a__filter(x1, x2, x3)] | = | x1 + x2 + x3 + 4 |
a__sieve(cons(s(N),Y)) | → | cons(s(mark(N)),sieve(filter(Y,N,N))) | (4) |
a__sieve(X) | → | sieve(X) | (15) |
mark(sieve(X)) | → | a__sieve(mark(X)) | (8) |
a__filter(cons(X,Y),0,M) | → | cons(0,filter(Y,M,M)) | (1) |
a__sieve(cons(0,Y)) | → | cons(0,sieve(Y)) | (3) |
a__nats(X) | → | nats(X) | (16) |
a__zprimes | → | zprimes | (17) |
a__nats(N) | → | cons(mark(N),nats(s(N))) | (5) |
mark(zprimes) | → | a__zprimes | (10) |
mark(filter(X1,X2,X3)) | → | a__filter(mark(X1),mark(X2),mark(X3)) | (7) |
a__filter(X1,X2,X3) | → | filter(X1,X2,X3) | (14) |
mark(0) | → | 0 | (12) |
mark(cons(X1,X2)) | → | cons(mark(X1),X2) | (11) |
mark(nats(X)) | → | a__nats(mark(X)) | (9) |
mark(s(X)) | → | s(mark(X)) | (13) |
a__zprimes | → | a__sieve(a__nats(s(s(0)))) | (6) |
a__filter(cons(X,Y),s(N),M) | → | cons(mark(X),filter(Y,N,M)) | (2) |
mark#(s(X)) | → | mark#(X) | (33) |
mark#(sieve(X)) | → | a__sieve#(mark(X)) | (24) |
mark#(sieve(X)) | → | mark#(X) | (32) |
mark#(zprimes) | → | a__zprimes# | (23) |
mark#(filter(X1,X2,X3)) | → | a__filter#(mark(X1),mark(X2),mark(X3)) | (31) |
a__filter#(cons(X,Y),s(N),M) | → | mark#(X) | (30) |
a__zprimes# | → | a__nats#(s(s(0))) | (29) |
a__sieve#(cons(s(N),Y)) | → | mark#(N) | (28) |
mark#(filter(X1,X2,X3)) | → | mark#(X3) | (22) |
mark#(filter(X1,X2,X3)) | → | mark#(X1) | (21) |
a__nats#(N) | → | mark#(N) | (20) |
mark#(nats(X)) | → | mark#(X) | (19) |
mark#(nats(X)) | → | a__nats#(mark(X)) | (27) |
mark#(filter(X1,X2,X3)) | → | mark#(X2) | (26) |
mark#(cons(X1,X2)) | → | mark#(X1) | (25) |
a__zprimes# | → | a__sieve#(a__nats(s(s(0)))) | (18) |
The dependency pairs are split into 0 components.