The rewrite relation of the following TRS is considered.
active(2nd(cons1(X,cons(Y,Z)))) | → | mark(Y) | (1) |
active(2nd(cons(X,X1))) | → | mark(2nd(cons1(X,X1))) | (2) |
active(from(X)) | → | mark(cons(X,from(s(X)))) | (3) |
mark(2nd(X)) | → | active(2nd(mark(X))) | (4) |
mark(cons1(X1,X2)) | → | active(cons1(mark(X1),mark(X2))) | (5) |
mark(cons(X1,X2)) | → | active(cons(mark(X1),X2)) | (6) |
mark(from(X)) | → | active(from(mark(X))) | (7) |
mark(s(X)) | → | active(s(mark(X))) | (8) |
2nd(mark(X)) | → | 2nd(X) | (9) |
2nd(active(X)) | → | 2nd(X) | (10) |
cons1(mark(X1),X2) | → | cons1(X1,X2) | (11) |
cons1(X1,mark(X2)) | → | cons1(X1,X2) | (12) |
cons1(active(X1),X2) | → | cons1(X1,X2) | (13) |
cons1(X1,active(X2)) | → | cons1(X1,X2) | (14) |
cons(mark(X1),X2) | → | cons(X1,X2) | (15) |
cons(X1,mark(X2)) | → | cons(X1,X2) | (16) |
cons(active(X1),X2) | → | cons(X1,X2) | (17) |
cons(X1,active(X2)) | → | cons(X1,X2) | (18) |
from(mark(X)) | → | from(X) | (19) |
from(active(X)) | → | from(X) | (20) |
s(mark(X)) | → | s(X) | (21) |
s(active(X)) | → | s(X) | (22) |
active#(2nd(cons1(X,cons(Y,Z)))) | → | mark#(Y) | (23) |
active#(2nd(cons(X,X1))) | → | cons1#(X,X1) | (24) |
active#(2nd(cons(X,X1))) | → | 2nd#(cons1(X,X1)) | (25) |
active#(2nd(cons(X,X1))) | → | mark#(2nd(cons1(X,X1))) | (26) |
active#(from(X)) | → | s#(X) | (27) |
active#(from(X)) | → | from#(s(X)) | (28) |
active#(from(X)) | → | cons#(X,from(s(X))) | (29) |
active#(from(X)) | → | mark#(cons(X,from(s(X)))) | (30) |
mark#(2nd(X)) | → | mark#(X) | (31) |
mark#(2nd(X)) | → | 2nd#(mark(X)) | (32) |
mark#(2nd(X)) | → | active#(2nd(mark(X))) | (33) |
mark#(cons1(X1,X2)) | → | mark#(X2) | (34) |
mark#(cons1(X1,X2)) | → | mark#(X1) | (35) |
mark#(cons1(X1,X2)) | → | cons1#(mark(X1),mark(X2)) | (36) |
mark#(cons1(X1,X2)) | → | active#(cons1(mark(X1),mark(X2))) | (37) |
mark#(cons(X1,X2)) | → | mark#(X1) | (38) |
mark#(cons(X1,X2)) | → | cons#(mark(X1),X2) | (39) |
mark#(cons(X1,X2)) | → | active#(cons(mark(X1),X2)) | (40) |
mark#(from(X)) | → | mark#(X) | (41) |
mark#(from(X)) | → | from#(mark(X)) | (42) |
mark#(from(X)) | → | active#(from(mark(X))) | (43) |
mark#(s(X)) | → | mark#(X) | (44) |
mark#(s(X)) | → | s#(mark(X)) | (45) |
mark#(s(X)) | → | active#(s(mark(X))) | (46) |
2nd#(mark(X)) | → | 2nd#(X) | (47) |
2nd#(active(X)) | → | 2nd#(X) | (48) |
cons1#(mark(X1),X2) | → | cons1#(X1,X2) | (49) |
cons1#(X1,mark(X2)) | → | cons1#(X1,X2) | (50) |
cons1#(active(X1),X2) | → | cons1#(X1,X2) | (51) |
cons1#(X1,active(X2)) | → | cons1#(X1,X2) | (52) |
cons#(mark(X1),X2) | → | cons#(X1,X2) | (53) |
cons#(X1,mark(X2)) | → | cons#(X1,X2) | (54) |
cons#(active(X1),X2) | → | cons#(X1,X2) | (55) |
cons#(X1,active(X2)) | → | cons#(X1,X2) | (56) |
from#(mark(X)) | → | from#(X) | (57) |
from#(active(X)) | → | from#(X) | (58) |
s#(mark(X)) | → | s#(X) | (59) |
s#(active(X)) | → | s#(X) | (60) |
The dependency pairs are split into 6 components.
mark#(s(X)) | → | mark#(X) | (44) |
mark#(from(X)) | → | active#(from(mark(X))) | (43) |
active#(from(X)) | → | mark#(cons(X,from(s(X)))) | (30) |
mark#(cons(X1,X2)) | → | mark#(X1) | (38) |
mark#(from(X)) | → | mark#(X) | (41) |
mark#(cons1(X1,X2)) | → | mark#(X1) | (35) |
mark#(cons1(X1,X2)) | → | mark#(X2) | (34) |
mark#(2nd(X)) | → | active#(2nd(mark(X))) | (33) |
active#(2nd(cons(X,X1))) | → | mark#(2nd(cons1(X,X1))) | (26) |
mark#(2nd(X)) | → | mark#(X) | (31) |
active#(2nd(cons1(X,cons(Y,Z)))) | → | mark#(Y) | (23) |
[mark(x1)] | = | 0 · x1 + -∞ |
[mark#(x1)] | = | 0 · x1 + -∞ |
[cons1(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[active(x1)] | = | 0 · x1 + -∞ |
[from(x1)] | = | 0 · x1 + 3 |
[cons(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[active#(x1)] | = | 0 · x1 + 0 |
[s(x1)] | = | 0 · x1 + 3 |
[2nd(x1)] | = | 1 · x1 + 0 |
active(2nd(cons1(X,cons(Y,Z)))) | → | mark(Y) | (1) |
active(2nd(cons(X,X1))) | → | mark(2nd(cons1(X,X1))) | (2) |
active(from(X)) | → | mark(cons(X,from(s(X)))) | (3) |
mark(2nd(X)) | → | active(2nd(mark(X))) | (4) |
mark(cons1(X1,X2)) | → | active(cons1(mark(X1),mark(X2))) | (5) |
mark(cons(X1,X2)) | → | active(cons(mark(X1),X2)) | (6) |
mark(from(X)) | → | active(from(mark(X))) | (7) |
mark(s(X)) | → | active(s(mark(X))) | (8) |
2nd(mark(X)) | → | 2nd(X) | (9) |
2nd(active(X)) | → | 2nd(X) | (10) |
cons1(mark(X1),X2) | → | cons1(X1,X2) | (11) |
cons1(X1,mark(X2)) | → | cons1(X1,X2) | (12) |
cons1(active(X1),X2) | → | cons1(X1,X2) | (13) |
cons1(X1,active(X2)) | → | cons1(X1,X2) | (14) |
cons(mark(X1),X2) | → | cons(X1,X2) | (15) |
cons(X1,mark(X2)) | → | cons(X1,X2) | (16) |
cons(active(X1),X2) | → | cons(X1,X2) | (17) |
cons(X1,active(X2)) | → | cons(X1,X2) | (18) |
from(mark(X)) | → | from(X) | (19) |
from(active(X)) | → | from(X) | (20) |
s(mark(X)) | → | s(X) | (21) |
s(active(X)) | → | s(X) | (22) |
mark#(2nd(X)) | → | mark#(X) | (31) |
active#(2nd(cons1(X,cons(Y,Z)))) | → | mark#(Y) | (23) |
The dependency pairs are split into 2 components.
mark#(s(X)) | → | mark#(X) | (44) |
mark#(from(X)) | → | active#(from(mark(X))) | (43) |
active#(from(X)) | → | mark#(cons(X,from(s(X)))) | (30) |
mark#(cons(X1,X2)) | → | mark#(X1) | (38) |
mark#(from(X)) | → | mark#(X) | (41) |
mark#(cons1(X1,X2)) | → | mark#(X1) | (35) |
mark#(cons1(X1,X2)) | → | mark#(X2) | (34) |
[mark(x1)] | = | 0 · x1 + 0 |
[mark#(x1)] | = | 0 · x1 + -∞ |
[cons1(x1, x2)] | = | 0 · x1 + 0 · x2 + -∞ |
[active(x1)] | = | 0 · x1 + 0 |
[from(x1)] | = | 5 · x1 + 7 |
[cons(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[active#(x1)] | = | 0 · x1 + 0 |
[s(x1)] | = | 0 · x1 + 1 |
[2nd(x1)] | = | 0 · x1 + 0 |
active(2nd(cons1(X,cons(Y,Z)))) | → | mark(Y) | (1) |
active(2nd(cons(X,X1))) | → | mark(2nd(cons1(X,X1))) | (2) |
active(from(X)) | → | mark(cons(X,from(s(X)))) | (3) |
mark(2nd(X)) | → | active(2nd(mark(X))) | (4) |
mark(cons1(X1,X2)) | → | active(cons1(mark(X1),mark(X2))) | (5) |
mark(cons(X1,X2)) | → | active(cons(mark(X1),X2)) | (6) |
mark(from(X)) | → | active(from(mark(X))) | (7) |
mark(s(X)) | → | active(s(mark(X))) | (8) |
2nd(mark(X)) | → | 2nd(X) | (9) |
2nd(active(X)) | → | 2nd(X) | (10) |
cons1(mark(X1),X2) | → | cons1(X1,X2) | (11) |
cons1(X1,mark(X2)) | → | cons1(X1,X2) | (12) |
cons1(active(X1),X2) | → | cons1(X1,X2) | (13) |
cons1(X1,active(X2)) | → | cons1(X1,X2) | (14) |
cons(mark(X1),X2) | → | cons(X1,X2) | (15) |
cons(X1,mark(X2)) | → | cons(X1,X2) | (16) |
cons(active(X1),X2) | → | cons(X1,X2) | (17) |
cons(X1,active(X2)) | → | cons(X1,X2) | (18) |
from(mark(X)) | → | from(X) | (19) |
from(active(X)) | → | from(X) | (20) |
s(mark(X)) | → | s(X) | (21) |
s(active(X)) | → | s(X) | (22) |
mark#(from(X)) | → | mark#(X) | (41) |
[mark(x1)] | = | 0 · x1 + -∞ |
[mark#(x1)] | = | 0 · x1 + -∞ |
[cons1(x1, x2)] | = | 4 · x1 + 0 · x2 + 0 |
[active(x1)] | = | 0 · x1 + -∞ |
[from(x1)] | = | 4 · x1 + 0 |
[cons(x1, x2)] | = | 4 · x1 + 0 · x2 + 0 |
[active#(x1)] | = | 0 · x1 + 0 |
[s(x1)] | = | 0 · x1 + -∞ |
[2nd(x1)] | = | 0 · x1 + -∞ |
active(2nd(cons1(X,cons(Y,Z)))) | → | mark(Y) | (1) |
active(2nd(cons(X,X1))) | → | mark(2nd(cons1(X,X1))) | (2) |
active(from(X)) | → | mark(cons(X,from(s(X)))) | (3) |
mark(2nd(X)) | → | active(2nd(mark(X))) | (4) |
mark(cons1(X1,X2)) | → | active(cons1(mark(X1),mark(X2))) | (5) |
mark(cons(X1,X2)) | → | active(cons(mark(X1),X2)) | (6) |
mark(from(X)) | → | active(from(mark(X))) | (7) |
mark(s(X)) | → | active(s(mark(X))) | (8) |
2nd(mark(X)) | → | 2nd(X) | (9) |
2nd(active(X)) | → | 2nd(X) | (10) |
cons1(mark(X1),X2) | → | cons1(X1,X2) | (11) |
cons1(X1,mark(X2)) | → | cons1(X1,X2) | (12) |
cons1(active(X1),X2) | → | cons1(X1,X2) | (13) |
cons1(X1,active(X2)) | → | cons1(X1,X2) | (14) |
cons(mark(X1),X2) | → | cons(X1,X2) | (15) |
cons(X1,mark(X2)) | → | cons(X1,X2) | (16) |
cons(active(X1),X2) | → | cons(X1,X2) | (17) |
cons(X1,active(X2)) | → | cons(X1,X2) | (18) |
from(mark(X)) | → | from(X) | (19) |
from(active(X)) | → | from(X) | (20) |
s(mark(X)) | → | s(X) | (21) |
s(active(X)) | → | s(X) | (22) |
mark#(cons(X1,X2)) | → | mark#(X1) | (38) |
mark#(cons1(X1,X2)) | → | mark#(X1) | (35) |
The dependency pairs are split into 1 component.
mark#(s(X)) | → | mark#(X) | (44) |
mark#(cons1(X1,X2)) | → | mark#(X2) | (34) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
mark#(s(X)) | → | mark#(X) | (44) |
1 | > | 1 | |
mark#(cons1(X1,X2)) | → | mark#(X2) | (34) |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
mark#(2nd(X)) | → | active#(2nd(mark(X))) | (33) |
active#(2nd(cons(X,X1))) | → | mark#(2nd(cons1(X,X1))) | (26) |
[mark(x1)] | = | 0 · x1 + 0 |
[mark#(x1)] | = | 0 · x1 + 0 |
[cons1(x1, x2)] | = | 0 · x1 + -3 · x2 + 4 |
[active(x1)] | = | 0 · x1 + 0 |
[from(x1)] | = | 3 · x1 + 6 |
[cons(x1, x2)] | = | 3 · x1 + 0 · x2 + 5 |
[active#(x1)] | = | 0 · x1 + 0 |
[s(x1)] | = | 0 · x1 + -5 |
[2nd(x1)] | = | 0 · x1 + -∞ |
active(2nd(cons1(X,cons(Y,Z)))) | → | mark(Y) | (1) |
active(2nd(cons(X,X1))) | → | mark(2nd(cons1(X,X1))) | (2) |
active(from(X)) | → | mark(cons(X,from(s(X)))) | (3) |
mark(2nd(X)) | → | active(2nd(mark(X))) | (4) |
mark(cons1(X1,X2)) | → | active(cons1(mark(X1),mark(X2))) | (5) |
mark(cons(X1,X2)) | → | active(cons(mark(X1),X2)) | (6) |
mark(from(X)) | → | active(from(mark(X))) | (7) |
mark(s(X)) | → | active(s(mark(X))) | (8) |
2nd(mark(X)) | → | 2nd(X) | (9) |
2nd(active(X)) | → | 2nd(X) | (10) |
cons1(mark(X1),X2) | → | cons1(X1,X2) | (11) |
cons1(X1,mark(X2)) | → | cons1(X1,X2) | (12) |
cons1(active(X1),X2) | → | cons1(X1,X2) | (13) |
cons1(X1,active(X2)) | → | cons1(X1,X2) | (14) |
cons(mark(X1),X2) | → | cons(X1,X2) | (15) |
cons(X1,mark(X2)) | → | cons(X1,X2) | (16) |
cons(active(X1),X2) | → | cons(X1,X2) | (17) |
cons(X1,active(X2)) | → | cons(X1,X2) | (18) |
from(mark(X)) | → | from(X) | (19) |
from(active(X)) | → | from(X) | (20) |
s(mark(X)) | → | s(X) | (21) |
s(active(X)) | → | s(X) | (22) |
active#(2nd(cons(X,X1))) | → | mark#(2nd(cons1(X,X1))) | (26) |
The dependency pairs are split into 0 components.
cons1#(mark(X1),X2) | → | cons1#(X1,X2) | (49) |
cons1#(X1,active(X2)) | → | cons1#(X1,X2) | (52) |
cons1#(active(X1),X2) | → | cons1#(X1,X2) | (51) |
cons1#(X1,mark(X2)) | → | cons1#(X1,X2) | (50) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
cons1#(mark(X1),X2) | → | cons1#(X1,X2) | (49) |
2 | ≥ | 2 | |
1 | > | 1 | |
cons1#(X1,active(X2)) | → | cons1#(X1,X2) | (52) |
2 | > | 2 | |
1 | ≥ | 1 | |
cons1#(active(X1),X2) | → | cons1#(X1,X2) | (51) |
2 | ≥ | 2 | |
1 | > | 1 | |
cons1#(X1,mark(X2)) | → | cons1#(X1,X2) | (50) |
2 | > | 2 | |
1 | ≥ | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
2nd#(mark(X)) | → | 2nd#(X) | (47) |
2nd#(active(X)) | → | 2nd#(X) | (48) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
2nd#(mark(X)) | → | 2nd#(X) | (47) |
1 | > | 1 | |
2nd#(active(X)) | → | 2nd#(X) | (48) |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
s#(mark(X)) | → | s#(X) | (59) |
s#(active(X)) | → | s#(X) | (60) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
s#(mark(X)) | → | s#(X) | (59) |
1 | > | 1 | |
s#(active(X)) | → | s#(X) | (60) |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
from#(mark(X)) | → | from#(X) | (57) |
from#(active(X)) | → | from#(X) | (58) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
from#(mark(X)) | → | from#(X) | (57) |
1 | > | 1 | |
from#(active(X)) | → | from#(X) | (58) |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
cons#(mark(X1),X2) | → | cons#(X1,X2) | (53) |
cons#(X1,active(X2)) | → | cons#(X1,X2) | (56) |
cons#(active(X1),X2) | → | cons#(X1,X2) | (55) |
cons#(X1,mark(X2)) | → | cons#(X1,X2) | (54) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
cons#(mark(X1),X2) | → | cons#(X1,X2) | (53) |
2 | ≥ | 2 | |
1 | > | 1 | |
cons#(X1,active(X2)) | → | cons#(X1,X2) | (56) |
2 | > | 2 | |
1 | ≥ | 1 | |
cons#(active(X1),X2) | → | cons#(X1,X2) | (55) |
2 | ≥ | 2 | |
1 | > | 1 | |
cons#(X1,mark(X2)) | → | cons#(X1,X2) | (54) |
2 | > | 2 | |
1 | ≥ | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.