The rewrite relation of the following TRS is considered.
active(fst(0,Z)) | → | mark(nil) | (1) |
active(fst(s(X),cons(Y,Z))) | → | mark(cons(Y,fst(X,Z))) | (2) |
active(from(X)) | → | mark(cons(X,from(s(X)))) | (3) |
active(add(0,X)) | → | mark(X) | (4) |
active(add(s(X),Y)) | → | mark(s(add(X,Y))) | (5) |
active(len(nil)) | → | mark(0) | (6) |
active(len(cons(X,Z))) | → | mark(s(len(Z))) | (7) |
mark(fst(X1,X2)) | → | active(fst(mark(X1),mark(X2))) | (8) |
mark(0) | → | active(0) | (9) |
mark(nil) | → | active(nil) | (10) |
mark(s(X)) | → | active(s(X)) | (11) |
mark(cons(X1,X2)) | → | active(cons(mark(X1),X2)) | (12) |
mark(from(X)) | → | active(from(mark(X))) | (13) |
mark(add(X1,X2)) | → | active(add(mark(X1),mark(X2))) | (14) |
mark(len(X)) | → | active(len(mark(X))) | (15) |
fst(mark(X1),X2) | → | fst(X1,X2) | (16) |
fst(X1,mark(X2)) | → | fst(X1,X2) | (17) |
fst(active(X1),X2) | → | fst(X1,X2) | (18) |
fst(X1,active(X2)) | → | fst(X1,X2) | (19) |
s(mark(X)) | → | s(X) | (20) |
s(active(X)) | → | s(X) | (21) |
cons(mark(X1),X2) | → | cons(X1,X2) | (22) |
cons(X1,mark(X2)) | → | cons(X1,X2) | (23) |
cons(active(X1),X2) | → | cons(X1,X2) | (24) |
cons(X1,active(X2)) | → | cons(X1,X2) | (25) |
from(mark(X)) | → | from(X) | (26) |
from(active(X)) | → | from(X) | (27) |
add(mark(X1),X2) | → | add(X1,X2) | (28) |
add(X1,mark(X2)) | → | add(X1,X2) | (29) |
add(active(X1),X2) | → | add(X1,X2) | (30) |
add(X1,active(X2)) | → | add(X1,X2) | (31) |
len(mark(X)) | → | len(X) | (32) |
len(active(X)) | → | len(X) | (33) |
active#(fst(0,Z)) | → | mark#(nil) | (34) |
active#(fst(s(X),cons(Y,Z))) | → | mark#(cons(Y,fst(X,Z))) | (35) |
active#(fst(s(X),cons(Y,Z))) | → | cons#(Y,fst(X,Z)) | (36) |
active#(fst(s(X),cons(Y,Z))) | → | fst#(X,Z) | (37) |
active#(from(X)) | → | mark#(cons(X,from(s(X)))) | (38) |
active#(from(X)) | → | cons#(X,from(s(X))) | (39) |
active#(from(X)) | → | from#(s(X)) | (40) |
active#(from(X)) | → | s#(X) | (41) |
active#(add(0,X)) | → | mark#(X) | (42) |
active#(add(s(X),Y)) | → | mark#(s(add(X,Y))) | (43) |
active#(add(s(X),Y)) | → | s#(add(X,Y)) | (44) |
active#(add(s(X),Y)) | → | add#(X,Y) | (45) |
active#(len(nil)) | → | mark#(0) | (46) |
active#(len(cons(X,Z))) | → | mark#(s(len(Z))) | (47) |
active#(len(cons(X,Z))) | → | s#(len(Z)) | (48) |
active#(len(cons(X,Z))) | → | len#(Z) | (49) |
mark#(fst(X1,X2)) | → | active#(fst(mark(X1),mark(X2))) | (50) |
mark#(fst(X1,X2)) | → | fst#(mark(X1),mark(X2)) | (51) |
mark#(fst(X1,X2)) | → | mark#(X1) | (52) |
mark#(fst(X1,X2)) | → | mark#(X2) | (53) |
mark#(0) | → | active#(0) | (54) |
mark#(nil) | → | active#(nil) | (55) |
mark#(s(X)) | → | active#(s(X)) | (56) |
mark#(cons(X1,X2)) | → | active#(cons(mark(X1),X2)) | (57) |
mark#(cons(X1,X2)) | → | cons#(mark(X1),X2) | (58) |
mark#(cons(X1,X2)) | → | mark#(X1) | (59) |
mark#(from(X)) | → | active#(from(mark(X))) | (60) |
mark#(from(X)) | → | from#(mark(X)) | (61) |
mark#(from(X)) | → | mark#(X) | (62) |
mark#(add(X1,X2)) | → | active#(add(mark(X1),mark(X2))) | (63) |
mark#(add(X1,X2)) | → | add#(mark(X1),mark(X2)) | (64) |
mark#(add(X1,X2)) | → | mark#(X1) | (65) |
mark#(add(X1,X2)) | → | mark#(X2) | (66) |
mark#(len(X)) | → | active#(len(mark(X))) | (67) |
mark#(len(X)) | → | len#(mark(X)) | (68) |
mark#(len(X)) | → | mark#(X) | (69) |
fst#(mark(X1),X2) | → | fst#(X1,X2) | (70) |
fst#(X1,mark(X2)) | → | fst#(X1,X2) | (71) |
fst#(active(X1),X2) | → | fst#(X1,X2) | (72) |
fst#(X1,active(X2)) | → | fst#(X1,X2) | (73) |
s#(mark(X)) | → | s#(X) | (74) |
s#(active(X)) | → | s#(X) | (75) |
cons#(mark(X1),X2) | → | cons#(X1,X2) | (76) |
cons#(X1,mark(X2)) | → | cons#(X1,X2) | (77) |
cons#(active(X1),X2) | → | cons#(X1,X2) | (78) |
cons#(X1,active(X2)) | → | cons#(X1,X2) | (79) |
from#(mark(X)) | → | from#(X) | (80) |
from#(active(X)) | → | from#(X) | (81) |
add#(mark(X1),X2) | → | add#(X1,X2) | (82) |
add#(X1,mark(X2)) | → | add#(X1,X2) | (83) |
add#(active(X1),X2) | → | add#(X1,X2) | (84) |
add#(X1,active(X2)) | → | add#(X1,X2) | (85) |
len#(mark(X)) | → | len#(X) | (86) |
len#(active(X)) | → | len#(X) | (87) |
The dependency pairs are split into 7 components.
active#(fst(s(X),cons(Y,Z))) | → | mark#(cons(Y,fst(X,Z))) | (35) |
mark#(cons(X1,X2)) | → | active#(cons(mark(X1),X2)) | (57) |
active#(from(X)) | → | mark#(cons(X,from(s(X)))) | (38) |
mark#(cons(X1,X2)) | → | mark#(X1) | (59) |
mark#(fst(X1,X2)) | → | active#(fst(mark(X1),mark(X2))) | (50) |
active#(add(0,X)) | → | mark#(X) | (42) |
mark#(fst(X1,X2)) | → | mark#(X1) | (52) |
mark#(fst(X1,X2)) | → | mark#(X2) | (53) |
mark#(from(X)) | → | active#(from(mark(X))) | (60) |
mark#(from(X)) | → | mark#(X) | (62) |
mark#(add(X1,X2)) | → | active#(add(mark(X1),mark(X2))) | (63) |
mark#(add(X1,X2)) | → | mark#(X1) | (65) |
mark#(add(X1,X2)) | → | mark#(X2) | (66) |
mark#(len(X)) | → | active#(len(mark(X))) | (67) |
mark#(len(X)) | → | mark#(X) | (69) |
We restrict the rewrite rules to the following usable rules of the DP problem.
active(fst(s(X),cons(Y,Z))) | → | mark(cons(Y,fst(X,Z))) | (2) |
mark(cons(X1,X2)) | → | active(cons(mark(X1),X2)) | (12) |
active(from(X)) | → | mark(cons(X,from(s(X)))) | (3) |
active(add(0,X)) | → | mark(X) | (4) |
mark(fst(X1,X2)) | → | active(fst(mark(X1),mark(X2))) | (8) |
mark(from(X)) | → | active(from(mark(X))) | (13) |
mark(add(X1,X2)) | → | active(add(mark(X1),mark(X2))) | (14) |
mark(len(X)) | → | active(len(mark(X))) | (15) |
mark(0) | → | active(0) | (9) |
mark(nil) | → | active(nil) | (10) |
mark(s(X)) | → | active(s(X)) | (11) |
len(active(X)) | → | len(X) | (33) |
len(mark(X)) | → | len(X) | (32) |
active(fst(0,Z)) | → | mark(nil) | (1) |
active(add(s(X),Y)) | → | mark(s(add(X,Y))) | (5) |
active(len(nil)) | → | mark(0) | (6) |
active(len(cons(X,Z))) | → | mark(s(len(Z))) | (7) |
cons(X1,mark(X2)) | → | cons(X1,X2) | (23) |
cons(mark(X1),X2) | → | cons(X1,X2) | (22) |
cons(active(X1),X2) | → | cons(X1,X2) | (24) |
cons(X1,active(X2)) | → | cons(X1,X2) | (25) |
fst(X1,mark(X2)) | → | fst(X1,X2) | (17) |
fst(mark(X1),X2) | → | fst(X1,X2) | (16) |
fst(active(X1),X2) | → | fst(X1,X2) | (18) |
fst(X1,active(X2)) | → | fst(X1,X2) | (19) |
from(active(X)) | → | from(X) | (27) |
from(mark(X)) | → | from(X) | (26) |
add(X1,mark(X2)) | → | add(X1,X2) | (29) |
add(mark(X1),X2) | → | add(X1,X2) | (28) |
add(active(X1),X2) | → | add(X1,X2) | (30) |
add(X1,active(X2)) | → | add(X1,X2) | (31) |
prec(fst) | = | 1 | weight(fst) | = | 3 | ||||
prec(s) | = | 0 | weight(s) | = | 3 | ||||
prec(add) | = | 2 | weight(add) | = | 1 | ||||
prec(0) | = | 3 | weight(0) | = | 6 | ||||
prec(nil) | = | 4 | weight(nil) | = | 8 |
π(active#) | = | 1 |
π(fst) | = | [1,2] |
π(s) | = | [] |
π(cons) | = | 1 |
π(mark#) | = | 1 |
π(mark) | = | 1 |
π(from) | = | 1 |
π(add) | = | [1,2] |
π(0) | = | [] |
π(len) | = | 1 |
π(active) | = | 1 |
π(nil) | = | [] |
active#(fst(s(X),cons(Y,Z))) | → | mark#(cons(Y,fst(X,Z))) | (35) |
active#(add(0,X)) | → | mark#(X) | (42) |
mark#(fst(X1,X2)) | → | mark#(X1) | (52) |
mark#(fst(X1,X2)) | → | mark#(X2) | (53) |
mark#(add(X1,X2)) | → | mark#(X1) | (65) |
mark#(add(X1,X2)) | → | mark#(X2) | (66) |
prec(add) | = | 1 | weight(add) | = | 3 | ||||
prec(len) | = | 3 | weight(len) | = | 4 | ||||
prec(0) | = | 2 | weight(0) | = | 3 | ||||
prec(nil) | = | 4 | weight(nil) | = | 1 | ||||
prec(s) | = | 0 | weight(s) | = | 3 |
π(mark#) | = | 1 |
π(cons) | = | 1 |
π(active#) | = | 1 |
π(mark) | = | 1 |
π(from) | = | 1 |
π(fst) | = | 2 |
π(add) | = | [2] |
π(len) | = | [1] |
π(active) | = | 1 |
π(0) | = | [] |
π(nil) | = | [] |
π(s) | = | [] |
mark#(len(X)) | → | mark#(X) | (69) |
prec(from) | = | 3 | weight(from) | = | 1 | ||||
prec(add) | = | 1 | weight(add) | = | 3 | ||||
prec(len) | = | 4 | weight(len) | = | 5 | ||||
prec(0) | = | 2 | weight(0) | = | 3 | ||||
prec(nil) | = | 5 | weight(nil) | = | 1 | ||||
prec(s) | = | 0 | weight(s) | = | 3 |
π(mark#) | = | 1 |
π(cons) | = | 1 |
π(active#) | = | 1 |
π(mark) | = | 1 |
π(from) | = | [1] |
π(fst) | = | 2 |
π(add) | = | [2] |
π(len) | = | [] |
π(active) | = | 1 |
π(0) | = | [] |
π(nil) | = | [] |
π(s) | = | [] |
active#(from(X)) | → | mark#(cons(X,from(s(X)))) | (38) |
mark#(from(X)) | → | mark#(X) | (62) |
The dependency pairs are split into 1 component.
mark#(cons(X1,X2)) | → | mark#(X1) | (59) |
We restrict the rewrite rules to the following usable rules of the DP problem.
There are no rules.
We restrict the innermost strategy to the following left hand sides.
cons(mark(x0),x1) |
cons(x0,mark(x1)) |
cons(active(x0),x1) |
cons(x0,active(x1)) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
mark#(cons(X1,X2)) | → | mark#(X1) | (59) |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
fst#(X1,mark(X2)) | → | fst#(X1,X2) | (71) |
fst#(mark(X1),X2) | → | fst#(X1,X2) | (70) |
fst#(active(X1),X2) | → | fst#(X1,X2) | (72) |
fst#(X1,active(X2)) | → | fst#(X1,X2) | (73) |
We restrict the rewrite rules to the following usable rules of the DP problem.
There are no rules.
We restrict the innermost strategy to the following left hand sides.
active(fst(0,x0)) |
active(fst(s(x0),cons(x1,x2))) |
active(from(x0)) |
active(add(0,x0)) |
active(add(s(x0),x1)) |
active(len(nil)) |
active(len(cons(x0,x1))) |
mark(fst(x0,x1)) |
mark(0) |
mark(nil) |
mark(s(x0)) |
mark(cons(x0,x1)) |
mark(from(x0)) |
mark(add(x0,x1)) |
mark(len(x0)) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
fst#(X1,mark(X2)) | → | fst#(X1,X2) | (71) |
1 | ≥ | 1 | |
2 | > | 2 | |
fst#(mark(X1),X2) | → | fst#(X1,X2) | (70) |
1 | > | 1 | |
2 | ≥ | 2 | |
fst#(active(X1),X2) | → | fst#(X1,X2) | (72) |
1 | > | 1 | |
2 | ≥ | 2 | |
fst#(X1,active(X2)) | → | fst#(X1,X2) | (73) |
1 | ≥ | 1 | |
2 | > | 2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
s#(active(X)) | → | s#(X) | (75) |
s#(mark(X)) | → | s#(X) | (74) |
We restrict the rewrite rules to the following usable rules of the DP problem.
There are no rules.
We restrict the innermost strategy to the following left hand sides.
active(fst(0,x0)) |
active(fst(s(x0),cons(x1,x2))) |
active(from(x0)) |
active(add(0,x0)) |
active(add(s(x0),x1)) |
active(len(nil)) |
active(len(cons(x0,x1))) |
mark(fst(x0,x1)) |
mark(0) |
mark(nil) |
mark(s(x0)) |
mark(cons(x0,x1)) |
mark(from(x0)) |
mark(add(x0,x1)) |
mark(len(x0)) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
s#(active(X)) | → | s#(X) | (75) |
1 | > | 1 | |
s#(mark(X)) | → | s#(X) | (74) |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
cons#(X1,mark(X2)) | → | cons#(X1,X2) | (77) |
cons#(mark(X1),X2) | → | cons#(X1,X2) | (76) |
cons#(active(X1),X2) | → | cons#(X1,X2) | (78) |
cons#(X1,active(X2)) | → | cons#(X1,X2) | (79) |
We restrict the rewrite rules to the following usable rules of the DP problem.
There are no rules.
We restrict the innermost strategy to the following left hand sides.
active(fst(0,x0)) |
active(fst(s(x0),cons(x1,x2))) |
active(from(x0)) |
active(add(0,x0)) |
active(add(s(x0),x1)) |
active(len(nil)) |
active(len(cons(x0,x1))) |
mark(fst(x0,x1)) |
mark(0) |
mark(nil) |
mark(s(x0)) |
mark(cons(x0,x1)) |
mark(from(x0)) |
mark(add(x0,x1)) |
mark(len(x0)) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
cons#(X1,mark(X2)) | → | cons#(X1,X2) | (77) |
1 | ≥ | 1 | |
2 | > | 2 | |
cons#(mark(X1),X2) | → | cons#(X1,X2) | (76) |
1 | > | 1 | |
2 | ≥ | 2 | |
cons#(active(X1),X2) | → | cons#(X1,X2) | (78) |
1 | > | 1 | |
2 | ≥ | 2 | |
cons#(X1,active(X2)) | → | cons#(X1,X2) | (79) |
1 | ≥ | 1 | |
2 | > | 2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
from#(active(X)) | → | from#(X) | (81) |
from#(mark(X)) | → | from#(X) | (80) |
We restrict the rewrite rules to the following usable rules of the DP problem.
There are no rules.
We restrict the innermost strategy to the following left hand sides.
active(fst(0,x0)) |
active(fst(s(x0),cons(x1,x2))) |
active(from(x0)) |
active(add(0,x0)) |
active(add(s(x0),x1)) |
active(len(nil)) |
active(len(cons(x0,x1))) |
mark(fst(x0,x1)) |
mark(0) |
mark(nil) |
mark(s(x0)) |
mark(cons(x0,x1)) |
mark(from(x0)) |
mark(add(x0,x1)) |
mark(len(x0)) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
from#(active(X)) | → | from#(X) | (81) |
1 | > | 1 | |
from#(mark(X)) | → | from#(X) | (80) |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
add#(X1,mark(X2)) | → | add#(X1,X2) | (83) |
add#(mark(X1),X2) | → | add#(X1,X2) | (82) |
add#(active(X1),X2) | → | add#(X1,X2) | (84) |
add#(X1,active(X2)) | → | add#(X1,X2) | (85) |
We restrict the rewrite rules to the following usable rules of the DP problem.
There are no rules.
We restrict the innermost strategy to the following left hand sides.
active(fst(0,x0)) |
active(fst(s(x0),cons(x1,x2))) |
active(from(x0)) |
active(add(0,x0)) |
active(add(s(x0),x1)) |
active(len(nil)) |
active(len(cons(x0,x1))) |
mark(fst(x0,x1)) |
mark(0) |
mark(nil) |
mark(s(x0)) |
mark(cons(x0,x1)) |
mark(from(x0)) |
mark(add(x0,x1)) |
mark(len(x0)) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
add#(X1,mark(X2)) | → | add#(X1,X2) | (83) |
1 | ≥ | 1 | |
2 | > | 2 | |
add#(mark(X1),X2) | → | add#(X1,X2) | (82) |
1 | > | 1 | |
2 | ≥ | 2 | |
add#(active(X1),X2) | → | add#(X1,X2) | (84) |
1 | > | 1 | |
2 | ≥ | 2 | |
add#(X1,active(X2)) | → | add#(X1,X2) | (85) |
1 | ≥ | 1 | |
2 | > | 2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
len#(active(X)) | → | len#(X) | (87) |
len#(mark(X)) | → | len#(X) | (86) |
We restrict the rewrite rules to the following usable rules of the DP problem.
There are no rules.
We restrict the innermost strategy to the following left hand sides.
active(fst(0,x0)) |
active(fst(s(x0),cons(x1,x2))) |
active(from(x0)) |
active(add(0,x0)) |
active(add(s(x0),x1)) |
active(len(nil)) |
active(len(cons(x0,x1))) |
mark(fst(x0,x1)) |
mark(0) |
mark(nil) |
mark(s(x0)) |
mark(cons(x0,x1)) |
mark(from(x0)) |
mark(add(x0,x1)) |
mark(len(x0)) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
len#(active(X)) | → | len#(X) | (87) |
1 | > | 1 | |
len#(mark(X)) | → | len#(X) | (86) |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.