The rewrite relation of the following TRS is considered.
active(nats) | → | mark(cons(0,incr(nats))) | (1) |
active(pairs) | → | mark(cons(0,incr(odds))) | (2) |
active(odds) | → | mark(incr(pairs)) | (3) |
active(incr(cons(X,XS))) | → | mark(cons(s(X),incr(XS))) | (4) |
active(head(cons(X,XS))) | → | mark(X) | (5) |
active(tail(cons(X,XS))) | → | mark(XS) | (6) |
mark(nats) | → | active(nats) | (7) |
mark(cons(X1,X2)) | → | active(cons(mark(X1),X2)) | (8) |
mark(0) | → | active(0) | (9) |
mark(incr(X)) | → | active(incr(mark(X))) | (10) |
mark(pairs) | → | active(pairs) | (11) |
mark(odds) | → | active(odds) | (12) |
mark(s(X)) | → | active(s(mark(X))) | (13) |
mark(head(X)) | → | active(head(mark(X))) | (14) |
mark(tail(X)) | → | active(tail(mark(X))) | (15) |
cons(mark(X1),X2) | → | cons(X1,X2) | (16) |
cons(X1,mark(X2)) | → | cons(X1,X2) | (17) |
cons(active(X1),X2) | → | cons(X1,X2) | (18) |
cons(X1,active(X2)) | → | cons(X1,X2) | (19) |
incr(mark(X)) | → | incr(X) | (20) |
incr(active(X)) | → | incr(X) | (21) |
s(mark(X)) | → | s(X) | (22) |
s(active(X)) | → | s(X) | (23) |
head(mark(X)) | → | head(X) | (24) |
head(active(X)) | → | head(X) | (25) |
tail(mark(X)) | → | tail(X) | (26) |
tail(active(X)) | → | tail(X) | (27) |
[cons(x1, x2)] | = |
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[s(x1)] | = |
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[active(x1)] | = |
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[incr(x1)] | = |
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[mark(x1)] | = |
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[tail(x1)] | = |
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[nats] | = |
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[odds] | = |
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[head(x1)] | = |
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[pairs] | = |
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[0] | = |
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active(head(cons(X,XS))) | → | mark(X) | (5) |
[cons(x1, x2)] | = |
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[s(x1)] | = |
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[active(x1)] | = |
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[incr(x1)] | = |
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[mark(x1)] | = |
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[tail(x1)] | = |
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[nats] | = |
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[odds] | = |
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[head(x1)] | = |
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[pairs] | = |
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[0] | = |
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active(tail(cons(X,XS))) | → | mark(XS) | (6) |
[cons(x1, x2)] | = |
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[s(x1)] | = |
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[active(x1)] | = |
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[incr(x1)] | = |
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[mark(x1)] | = |
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[tail(x1)] | = |
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[nats] | = |
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[odds] | = |
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[head(x1)] | = |
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[pairs] | = |
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[0] | = |
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mark(head(X)) | → | active(head(mark(X))) | (14) |
[cons(x1, x2)] | = |
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[s(x1)] | = |
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[active(x1)] | = |
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[incr(x1)] | = |
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[mark(x1)] | = |
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[tail(x1)] | = |
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[nats] | = |
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[odds] | = |
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[head(x1)] | = |
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[pairs] | = |
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[0] | = |
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head(mark(X)) | → | head(X) | (24) |
head(active(X)) | → | head(X) | (25) |
[cons(x1, x2)] | = |
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[s(x1)] | = |
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[active(x1)] | = |
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[incr(x1)] | = |
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[mark(x1)] | = |
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[tail(x1)] | = |
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[nats] | = |
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[odds] | = |
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[pairs] | = |
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[0] | = |
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cons(X1,mark(X2)) | → | cons(X1,X2) | (17) |
active#(nats) | → | incr#(nats) | (28) |
active#(nats) | → | cons#(0,incr(nats)) | (29) |
active#(nats) | → | mark#(cons(0,incr(nats))) | (30) |
active#(pairs) | → | incr#(odds) | (31) |
active#(pairs) | → | cons#(0,incr(odds)) | (32) |
active#(pairs) | → | mark#(cons(0,incr(odds))) | (33) |
active#(odds) | → | incr#(pairs) | (34) |
active#(odds) | → | mark#(incr(pairs)) | (35) |
active#(incr(cons(X,XS))) | → | incr#(XS) | (36) |
active#(incr(cons(X,XS))) | → | s#(X) | (37) |
active#(incr(cons(X,XS))) | → | cons#(s(X),incr(XS)) | (38) |
active#(incr(cons(X,XS))) | → | mark#(cons(s(X),incr(XS))) | (39) |
mark#(nats) | → | active#(nats) | (40) |
mark#(cons(X1,X2)) | → | mark#(X1) | (41) |
mark#(cons(X1,X2)) | → | cons#(mark(X1),X2) | (42) |
mark#(cons(X1,X2)) | → | active#(cons(mark(X1),X2)) | (43) |
mark#(0) | → | active#(0) | (44) |
mark#(incr(X)) | → | mark#(X) | (45) |
mark#(incr(X)) | → | incr#(mark(X)) | (46) |
mark#(incr(X)) | → | active#(incr(mark(X))) | (47) |
mark#(pairs) | → | active#(pairs) | (48) |
mark#(odds) | → | active#(odds) | (49) |
mark#(s(X)) | → | mark#(X) | (50) |
mark#(s(X)) | → | s#(mark(X)) | (51) |
mark#(s(X)) | → | active#(s(mark(X))) | (52) |
mark#(tail(X)) | → | mark#(X) | (53) |
mark#(tail(X)) | → | tail#(mark(X)) | (54) |
mark#(tail(X)) | → | active#(tail(mark(X))) | (55) |
cons#(mark(X1),X2) | → | cons#(X1,X2) | (56) |
cons#(active(X1),X2) | → | cons#(X1,X2) | (57) |
cons#(X1,active(X2)) | → | cons#(X1,X2) | (58) |
incr#(mark(X)) | → | incr#(X) | (59) |
incr#(active(X)) | → | incr#(X) | (60) |
s#(mark(X)) | → | s#(X) | (61) |
s#(active(X)) | → | s#(X) | (62) |
tail#(mark(X)) | → | tail#(X) | (63) |
tail#(active(X)) | → | tail#(X) | (64) |
The dependency pairs are split into 5 components.
mark#(tail(X)) | → | mark#(X) | (53) |
mark#(s(X)) | → | mark#(X) | (50) |
mark#(odds) | → | active#(odds) | (49) |
active#(odds) | → | mark#(incr(pairs)) | (35) |
mark#(incr(X)) | → | active#(incr(mark(X))) | (47) |
active#(incr(cons(X,XS))) | → | mark#(cons(s(X),incr(XS))) | (39) |
mark#(cons(X1,X2)) | → | mark#(X1) | (41) |
mark#(pairs) | → | active#(pairs) | (48) |
active#(pairs) | → | mark#(cons(0,incr(odds))) | (33) |
mark#(incr(X)) | → | mark#(X) | (45) |
mark#(nats) | → | active#(nats) | (40) |
active#(nats) | → | mark#(cons(0,incr(nats))) | (30) |
prec(mark#) | = | 0 | stat(mark#) | = | lex | |
prec(active#) | = | 0 | stat(active#) | = | lex | |
prec(tail) | = | 0 | stat(tail) | = | lex | |
prec(s) | = | 0 | stat(s) | = | lex | |
prec(odds) | = | 2 | stat(odds) | = | lex | |
prec(pairs) | = | 1 | stat(pairs) | = | lex | |
prec(mark) | = | 0 | stat(mark) | = | lex | |
prec(cons) | = | 0 | stat(cons) | = | lex | |
prec(incr) | = | 0 | stat(incr) | = | lex | |
prec(0) | = | 0 | stat(0) | = | lex | |
prec(active) | = | 0 | stat(active) | = | lex | |
prec(nats) | = | 1 | stat(nats) | = | lex |
π(mark#) | = | 1 |
π(active#) | = | 1 |
π(tail) | = | 1 |
π(s) | = | 1 |
π(odds) | = | [] |
π(pairs) | = | [] |
π(mark) | = | 1 |
π(cons) | = | 1 |
π(incr) | = | 1 |
π(0) | = | [] |
π(active) | = | 1 |
π(nats) | = | [] |
active(nats) | → | mark(cons(0,incr(nats))) | (1) |
active(pairs) | → | mark(cons(0,incr(odds))) | (2) |
active(odds) | → | mark(incr(pairs)) | (3) |
active(incr(cons(X,XS))) | → | mark(cons(s(X),incr(XS))) | (4) |
mark(nats) | → | active(nats) | (7) |
mark(cons(X1,X2)) | → | active(cons(mark(X1),X2)) | (8) |
mark(0) | → | active(0) | (9) |
mark(incr(X)) | → | active(incr(mark(X))) | (10) |
mark(pairs) | → | active(pairs) | (11) |
mark(odds) | → | active(odds) | (12) |
mark(s(X)) | → | active(s(mark(X))) | (13) |
mark(tail(X)) | → | active(tail(mark(X))) | (15) |
cons(mark(X1),X2) | → | cons(X1,X2) | (16) |
cons(active(X1),X2) | → | cons(X1,X2) | (18) |
cons(X1,active(X2)) | → | cons(X1,X2) | (19) |
incr(mark(X)) | → | incr(X) | (20) |
incr(active(X)) | → | incr(X) | (21) |
s(mark(X)) | → | s(X) | (22) |
s(active(X)) | → | s(X) | (23) |
tail(mark(X)) | → | tail(X) | (26) |
tail(active(X)) | → | tail(X) | (27) |
active#(odds) | → | mark#(incr(pairs)) | (35) |
active#(pairs) | → | mark#(cons(0,incr(odds))) | (33) |
active#(nats) | → | mark#(cons(0,incr(nats))) | (30) |
The dependency pairs are split into 1 component.
mark#(tail(X)) | → | mark#(X) | (53) |
mark#(s(X)) | → | mark#(X) | (50) |
mark#(incr(X)) | → | active#(incr(mark(X))) | (47) |
active#(incr(cons(X,XS))) | → | mark#(cons(s(X),incr(XS))) | (39) |
mark#(cons(X1,X2)) | → | mark#(X1) | (41) |
mark#(incr(X)) | → | mark#(X) | (45) |
prec(mark#) | = | 0 | stat(mark#) | = | lex | |
prec(active#) | = | 0 | stat(active#) | = | lex | |
prec(tail) | = | 0 | stat(tail) | = | lex | |
prec(s) | = | 0 | stat(s) | = | lex | |
prec(odds) | = | 2 | stat(odds) | = | lex | |
prec(pairs) | = | 1 | stat(pairs) | = | lex | |
prec(mark) | = | 0 | stat(mark) | = | lex | |
prec(cons) | = | 0 | stat(cons) | = | lex | |
prec(incr) | = | 0 | stat(incr) | = | lex | |
prec(0) | = | 0 | stat(0) | = | lex | |
prec(active) | = | 0 | stat(active) | = | lex | |
prec(nats) | = | 2 | stat(nats) | = | lex |
π(mark#) | = | 1 |
π(active#) | = | 1 |
π(tail) | = | [1] |
π(s) | = | 1 |
π(odds) | = | [] |
π(pairs) | = | [] |
π(mark) | = | 1 |
π(cons) | = | 1 |
π(incr) | = | 1 |
π(0) | = | [] |
π(active) | = | 1 |
π(nats) | = | [] |
active(nats) | → | mark(cons(0,incr(nats))) | (1) |
active(pairs) | → | mark(cons(0,incr(odds))) | (2) |
active(odds) | → | mark(incr(pairs)) | (3) |
active(incr(cons(X,XS))) | → | mark(cons(s(X),incr(XS))) | (4) |
mark(nats) | → | active(nats) | (7) |
mark(cons(X1,X2)) | → | active(cons(mark(X1),X2)) | (8) |
mark(0) | → | active(0) | (9) |
mark(incr(X)) | → | active(incr(mark(X))) | (10) |
mark(pairs) | → | active(pairs) | (11) |
mark(odds) | → | active(odds) | (12) |
mark(s(X)) | → | active(s(mark(X))) | (13) |
mark(tail(X)) | → | active(tail(mark(X))) | (15) |
cons(mark(X1),X2) | → | cons(X1,X2) | (16) |
cons(active(X1),X2) | → | cons(X1,X2) | (18) |
cons(X1,active(X2)) | → | cons(X1,X2) | (19) |
incr(mark(X)) | → | incr(X) | (20) |
incr(active(X)) | → | incr(X) | (21) |
s(mark(X)) | → | s(X) | (22) |
s(active(X)) | → | s(X) | (23) |
tail(mark(X)) | → | tail(X) | (26) |
tail(active(X)) | → | tail(X) | (27) |
mark#(tail(X)) | → | mark#(X) | (53) |
prec(mark#) | = | 0 | stat(mark#) | = | lex | |
prec(active#) | = | 0 | stat(active#) | = | lex | |
prec(tail) | = | 0 | stat(tail) | = | lex | |
prec(s) | = | 0 | stat(s) | = | lex | |
prec(odds) | = | 6 | stat(odds) | = | lex | |
prec(pairs) | = | 1 | stat(pairs) | = | lex | |
prec(mark) | = | 0 | stat(mark) | = | lex | |
prec(cons) | = | 0 | stat(cons) | = | lex | |
prec(incr) | = | 1 | stat(incr) | = | lex | |
prec(0) | = | 0 | stat(0) | = | lex | |
prec(active) | = | 0 | stat(active) | = | lex | |
prec(nats) | = | 1 | stat(nats) | = | lex |
π(mark#) | = | 1 |
π(active#) | = | 1 |
π(tail) | = | [] |
π(s) | = | 1 |
π(odds) | = | [] |
π(pairs) | = | [] |
π(mark) | = | 1 |
π(cons) | = | [1] |
π(incr) | = | [1] |
π(0) | = | [] |
π(active) | = | 1 |
π(nats) | = | [] |
active(nats) | → | mark(cons(0,incr(nats))) | (1) |
active(pairs) | → | mark(cons(0,incr(odds))) | (2) |
active(odds) | → | mark(incr(pairs)) | (3) |
active(incr(cons(X,XS))) | → | mark(cons(s(X),incr(XS))) | (4) |
mark(nats) | → | active(nats) | (7) |
mark(cons(X1,X2)) | → | active(cons(mark(X1),X2)) | (8) |
mark(0) | → | active(0) | (9) |
mark(incr(X)) | → | active(incr(mark(X))) | (10) |
mark(pairs) | → | active(pairs) | (11) |
mark(odds) | → | active(odds) | (12) |
mark(s(X)) | → | active(s(mark(X))) | (13) |
mark(tail(X)) | → | active(tail(mark(X))) | (15) |
cons(mark(X1),X2) | → | cons(X1,X2) | (16) |
cons(active(X1),X2) | → | cons(X1,X2) | (18) |
cons(X1,active(X2)) | → | cons(X1,X2) | (19) |
incr(mark(X)) | → | incr(X) | (20) |
incr(active(X)) | → | incr(X) | (21) |
s(mark(X)) | → | s(X) | (22) |
s(active(X)) | → | s(X) | (23) |
tail(mark(X)) | → | tail(X) | (26) |
tail(active(X)) | → | tail(X) | (27) |
active#(incr(cons(X,XS))) | → | mark#(cons(s(X),incr(XS))) | (39) |
mark#(cons(X1,X2)) | → | mark#(X1) | (41) |
mark#(incr(X)) | → | mark#(X) | (45) |
The dependency pairs are split into 1 component.
mark#(s(X)) | → | mark#(X) | (50) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
mark#(s(X)) | → | mark#(X) | (50) |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
incr#(active(X)) | → | incr#(X) | (60) |
incr#(mark(X)) | → | incr#(X) | (59) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
incr#(active(X)) | → | incr#(X) | (60) |
1 | > | 1 | |
incr#(mark(X)) | → | incr#(X) | (59) |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
cons#(X1,active(X2)) | → | cons#(X1,X2) | (58) |
cons#(active(X1),X2) | → | cons#(X1,X2) | (57) |
cons#(mark(X1),X2) | → | cons#(X1,X2) | (56) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
cons#(X1,active(X2)) | → | cons#(X1,X2) | (58) |
2 | > | 2 | |
1 | ≥ | 1 | |
cons#(active(X1),X2) | → | cons#(X1,X2) | (57) |
2 | ≥ | 2 | |
1 | > | 1 | |
cons#(mark(X1),X2) | → | cons#(X1,X2) | (56) |
2 | ≥ | 2 | |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
s#(mark(X)) | → | s#(X) | (61) |
s#(active(X)) | → | s#(X) | (62) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
s#(mark(X)) | → | s#(X) | (61) |
1 | > | 1 | |
s#(active(X)) | → | s#(X) | (62) |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
tail#(mark(X)) | → | tail#(X) | (63) |
tail#(active(X)) | → | tail#(X) | (64) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
tail#(mark(X)) | → | tail#(X) | (63) |
1 | > | 1 | |
tail#(active(X)) | → | tail#(X) | (64) |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.