The rewrite relation of the following TRS is considered.
a__from(X) | → | cons(mark(X),from(s(X))) | (1) |
a__after(0,XS) | → | mark(XS) | (2) |
a__after(s(N),cons(X,XS)) | → | a__after(mark(N),mark(XS)) | (3) |
mark(from(X)) | → | a__from(mark(X)) | (4) |
mark(after(X1,X2)) | → | a__after(mark(X1),mark(X2)) | (5) |
mark(cons(X1,X2)) | → | cons(mark(X1),X2) | (6) |
mark(s(X)) | → | s(mark(X)) | (7) |
mark(0) | → | 0 | (8) |
a__from(X) | → | from(X) | (9) |
a__after(X1,X2) | → | after(X1,X2) | (10) |
a__from#(X) | → | mark#(X) | (11) |
a__after#(0,XS) | → | mark#(XS) | (12) |
a__after#(s(N),cons(X,XS)) | → | mark#(XS) | (13) |
a__after#(s(N),cons(X,XS)) | → | mark#(N) | (14) |
a__after#(s(N),cons(X,XS)) | → | a__after#(mark(N),mark(XS)) | (15) |
mark#(from(X)) | → | mark#(X) | (16) |
mark#(from(X)) | → | a__from#(mark(X)) | (17) |
mark#(after(X1,X2)) | → | mark#(X2) | (18) |
mark#(after(X1,X2)) | → | mark#(X1) | (19) |
mark#(after(X1,X2)) | → | a__after#(mark(X1),mark(X2)) | (20) |
mark#(cons(X1,X2)) | → | mark#(X1) | (21) |
mark#(s(X)) | → | mark#(X) | (22) |
[cons(x1, x2)] | = | 0 · x1 + 0 · x2 + 3 |
[a__from#(x1)] | = | 0 · x1 + 3 |
[mark(x1)] | = | 0 · x1 + 0 |
[from(x1)] | = | 0 · x1 + 3 |
[0] | = | 0 |
[a__after#(x1, x2)] | = | 7 · x1 + 0 · x2 + 0 |
[a__from(x1)] | = | 0 · x1 + 3 |
[after(x1, x2)] | = | 7 · x1 + 0 · x2 + 7 |
[mark#(x1)] | = | 0 · x1 + -∞ |
[a__after(x1, x2)] | = | 7 · x1 + 0 · x2 + 7 |
[s(x1)] | = | 0 · x1 + 0 |
a__from(X) | → | cons(mark(X),from(s(X))) | (1) |
a__after(0,XS) | → | mark(XS) | (2) |
a__after(s(N),cons(X,XS)) | → | a__after(mark(N),mark(XS)) | (3) |
mark(from(X)) | → | a__from(mark(X)) | (4) |
mark(after(X1,X2)) | → | a__after(mark(X1),mark(X2)) | (5) |
mark(cons(X1,X2)) | → | cons(mark(X1),X2) | (6) |
mark(s(X)) | → | s(mark(X)) | (7) |
mark(0) | → | 0 | (8) |
a__from(X) | → | from(X) | (9) |
a__after(X1,X2) | → | after(X1,X2) | (10) |
a__after#(s(N),cons(X,XS)) | → | mark#(N) | (14) |
mark#(after(X1,X2)) | → | mark#(X1) | (19) |
[cons(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[a__from#(x1)] | = | 1 · x1 + 0 |
[mark(x1)] | = | 0 · x1 + 0 |
[from(x1)] | = | 1 · x1 + 1 |
[0] | = | 0 |
[a__after#(x1, x2)] | = | -∞ · x1 + 0 · x2 + -∞ |
[a__from(x1)] | = | 1 · x1 + 1 |
[after(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[mark#(x1)] | = | 0 · x1 + -∞ |
[a__after(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[s(x1)] | = | 0 · x1 + -∞ |
a__from(X) | → | cons(mark(X),from(s(X))) | (1) |
a__after(0,XS) | → | mark(XS) | (2) |
a__after(s(N),cons(X,XS)) | → | a__after(mark(N),mark(XS)) | (3) |
mark(from(X)) | → | a__from(mark(X)) | (4) |
mark(after(X1,X2)) | → | a__after(mark(X1),mark(X2)) | (5) |
mark(cons(X1,X2)) | → | cons(mark(X1),X2) | (6) |
mark(s(X)) | → | s(mark(X)) | (7) |
mark(0) | → | 0 | (8) |
a__from(X) | → | from(X) | (9) |
a__after(X1,X2) | → | after(X1,X2) | (10) |
a__from#(X) | → | mark#(X) | (11) |
mark#(from(X)) | → | mark#(X) | (16) |
The dependency pairs are split into 1 component.
a__after#(0,XS) | → | mark#(XS) | (12) |
mark#(after(X1,X2)) | → | mark#(X2) | (18) |
mark#(after(X1,X2)) | → | a__after#(mark(X1),mark(X2)) | (20) |
a__after#(s(N),cons(X,XS)) | → | mark#(XS) | (13) |
mark#(cons(X1,X2)) | → | mark#(X1) | (21) |
mark#(s(X)) | → | mark#(X) | (22) |
a__after#(s(N),cons(X,XS)) | → | a__after#(mark(N),mark(XS)) | (15) |
[cons(x1, x2)] | = | 0 · x1 + 0 · x2 + -∞ |
[mark(x1)] | = | 0 · x1 + 0 |
[from(x1)] | = | 0 · x1 + -∞ |
[0] | = | 3 |
[a__after#(x1, x2)] | = | -∞ · x1 + 0 · x2 + 0 |
[a__from(x1)] | = | 0 · x1 + 0 |
[after(x1, x2)] | = | 0 · x1 + 2 · x2 + 5 |
[mark#(x1)] | = | 0 · x1 + 0 |
[a__after(x1, x2)] | = | 0 · x1 + 2 · x2 + 5 |
[s(x1)] | = | 0 · x1 + -∞ |
a__from(X) | → | cons(mark(X),from(s(X))) | (1) |
a__after(0,XS) | → | mark(XS) | (2) |
a__after(s(N),cons(X,XS)) | → | a__after(mark(N),mark(XS)) | (3) |
mark(from(X)) | → | a__from(mark(X)) | (4) |
mark(after(X1,X2)) | → | a__after(mark(X1),mark(X2)) | (5) |
mark(cons(X1,X2)) | → | cons(mark(X1),X2) | (6) |
mark(s(X)) | → | s(mark(X)) | (7) |
mark(0) | → | 0 | (8) |
a__from(X) | → | from(X) | (9) |
a__after(X1,X2) | → | after(X1,X2) | (10) |
mark#(after(X1,X2)) | → | mark#(X2) | (18) |
mark#(after(X1,X2)) | → | a__after#(mark(X1),mark(X2)) | (20) |
The dependency pairs are split into 2 components.
a__after#(s(N),cons(X,XS)) | → | a__after#(mark(N),mark(XS)) | (15) |
[cons(x1, x2)] | = | -∞ · x1 + 0 · x2 + 1 |
[mark(x1)] | = | 0 · x1 + 2 |
[from(x1)] | = | -∞ · x1 + 0 |
[0] | = | 0 |
[a__after#(x1, x2)] | = | 0 · x1 + -∞ · x2 + 3 |
[a__from(x1)] | = | -∞ · x1 + 1 |
[after(x1, x2)] | = | -∞ · x1 + 0 · x2 + 4 |
[a__after(x1, x2)] | = | -∞ · x1 + 0 · x2 + 4 |
[s(x1)] | = | 2 · x1 + 4 |
a__from(X) | → | cons(mark(X),from(s(X))) | (1) |
a__after(0,XS) | → | mark(XS) | (2) |
a__after(s(N),cons(X,XS)) | → | a__after(mark(N),mark(XS)) | (3) |
mark(from(X)) | → | a__from(mark(X)) | (4) |
mark(after(X1,X2)) | → | a__after(mark(X1),mark(X2)) | (5) |
mark(cons(X1,X2)) | → | cons(mark(X1),X2) | (6) |
mark(s(X)) | → | s(mark(X)) | (7) |
mark(0) | → | 0 | (8) |
a__from(X) | → | from(X) | (9) |
a__after(X1,X2) | → | after(X1,X2) | (10) |
a__after#(s(N),cons(X,XS)) | → | a__after#(mark(N),mark(XS)) | (15) |
There are no pairs anymore.
mark#(cons(X1,X2)) | → | mark#(X1) | (21) |
mark#(s(X)) | → | mark#(X) | (22) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
mark#(cons(X1,X2)) | → | mark#(X1) | (21) |
1 | > | 1 | |
mark#(s(X)) | → | mark#(X) | (22) |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.