The rewrite relation of the following TRS is considered.
a__fst(0,Z) | → | nil | (1) |
a__fst(s(X),cons(Y,Z)) | → | cons(mark(Y),fst(X,Z)) | (2) |
a__from(X) | → | cons(mark(X),from(s(X))) | (3) |
a__add(0,X) | → | mark(X) | (4) |
a__add(s(X),Y) | → | s(add(X,Y)) | (5) |
a__len(nil) | → | 0 | (6) |
a__len(cons(X,Z)) | → | s(len(Z)) | (7) |
mark(fst(X1,X2)) | → | a__fst(mark(X1),mark(X2)) | (8) |
mark(from(X)) | → | a__from(mark(X)) | (9) |
mark(add(X1,X2)) | → | a__add(mark(X1),mark(X2)) | (10) |
mark(len(X)) | → | a__len(mark(X)) | (11) |
mark(0) | → | 0 | (12) |
mark(s(X)) | → | s(X) | (13) |
mark(nil) | → | nil | (14) |
mark(cons(X1,X2)) | → | cons(mark(X1),X2) | (15) |
a__fst(X1,X2) | → | fst(X1,X2) | (16) |
a__from(X) | → | from(X) | (17) |
a__add(X1,X2) | → | add(X1,X2) | (18) |
a__len(X) | → | len(X) | (19) |
a__fst#(s(X),cons(Y,Z)) | → | mark#(Y) | (20) |
a__from#(X) | → | mark#(X) | (21) |
a__add#(0,X) | → | mark#(X) | (22) |
mark#(fst(X1,X2)) | → | a__fst#(mark(X1),mark(X2)) | (23) |
mark#(fst(X1,X2)) | → | mark#(X1) | (24) |
mark#(fst(X1,X2)) | → | mark#(X2) | (25) |
mark#(from(X)) | → | a__from#(mark(X)) | (26) |
mark#(from(X)) | → | mark#(X) | (27) |
mark#(add(X1,X2)) | → | a__add#(mark(X1),mark(X2)) | (28) |
mark#(add(X1,X2)) | → | mark#(X1) | (29) |
mark#(add(X1,X2)) | → | mark#(X2) | (30) |
mark#(len(X)) | → | a__len#(mark(X)) | (31) |
mark#(len(X)) | → | mark#(X) | (32) |
mark#(cons(X1,X2)) | → | mark#(X1) | (33) |
The dependency pairs are split into 1 component.
mark#(fst(X1,X2)) | → | a__fst#(mark(X1),mark(X2)) | (23) |
a__fst#(s(X),cons(Y,Z)) | → | mark#(Y) | (20) |
mark#(fst(X1,X2)) | → | mark#(X1) | (24) |
mark#(fst(X1,X2)) | → | mark#(X2) | (25) |
mark#(from(X)) | → | a__from#(mark(X)) | (26) |
a__from#(X) | → | mark#(X) | (21) |
mark#(from(X)) | → | mark#(X) | (27) |
mark#(add(X1,X2)) | → | a__add#(mark(X1),mark(X2)) | (28) |
a__add#(0,X) | → | mark#(X) | (22) |
mark#(add(X1,X2)) | → | mark#(X1) | (29) |
mark#(add(X1,X2)) | → | mark#(X2) | (30) |
mark#(len(X)) | → | mark#(X) | (32) |
mark#(cons(X1,X2)) | → | mark#(X1) | (33) |
prec(mark#) | = | 0 | stat(mark#) | = | mul | |
prec(fst) | = | 3 | stat(fst) | = | mul | |
prec(a__fst#) | = | 0 | stat(a__fst#) | = | mul | |
prec(mark) | = | 3 | stat(mark) | = | mul | |
prec(s) | = | 1 | stat(s) | = | mul | |
prec(from) | = | 3 | stat(from) | = | mul | |
prec(a__from#) | = | 0 | stat(a__from#) | = | mul | |
prec(add) | = | 3 | stat(add) | = | mul | |
prec(a__add#) | = | 0 | stat(a__add#) | = | mul | |
prec(0) | = | 3 | stat(0) | = | mul | |
prec(len) | = | 3 | stat(len) | = | mul | |
prec(a__fst) | = | 3 | stat(a__fst) | = | mul | |
prec(a__from) | = | 3 | stat(a__from) | = | mul | |
prec(a__add) | = | 3 | stat(a__add) | = | mul | |
prec(a__len) | = | 3 | stat(a__len) | = | mul | |
prec(nil) | = | 2 | stat(nil) | = | mul |
π(mark#) | = | [1] |
π(fst) | = | [1,2] |
π(a__fst#) | = | [2] |
π(mark) | = | [1] |
π(s) | = | [] |
π(cons) | = | 1 |
π(from) | = | [1] |
π(a__from#) | = | [1] |
π(add) | = | [1,2] |
π(a__add#) | = | [2] |
π(0) | = | [] |
π(len) | = | [1] |
π(a__fst) | = | [1,2] |
π(a__from) | = | [1] |
π(a__add) | = | [1,2] |
π(a__len) | = | [1] |
π(nil) | = | [] |
mark#(fst(X1,X2)) | → | a__fst#(mark(X1),mark(X2)) | (23) |
mark#(fst(X1,X2)) | → | mark#(X1) | (24) |
mark#(fst(X1,X2)) | → | mark#(X2) | (25) |
mark#(from(X)) | → | mark#(X) | (27) |
mark#(add(X1,X2)) | → | a__add#(mark(X1),mark(X2)) | (28) |
mark#(add(X1,X2)) | → | mark#(X1) | (29) |
mark#(add(X1,X2)) | → | mark#(X2) | (30) |
mark#(len(X)) | → | mark#(X) | (32) |
The dependency pairs are split into 1 component.
a__from#(X) | → | mark#(X) | (21) |
mark#(from(X)) | → | a__from#(mark(X)) | (26) |
mark#(cons(X1,X2)) | → | mark#(X1) | (33) |
[a__from#(x1)] | = | 2 · x1 |
[mark(x1)] | = | x1 |
[fst(x1, x2)] | = | 2 · x2 |
[a__fst(x1, x2)] | = | 2 · x2 |
[from(x1)] | = | 2 + 2 · x1 |
[a__from(x1)] | = | 2 + 2 · x1 |
[add(x1, x2)] | = | 2 + x1 + x2 |
[a__add(x1, x2)] | = | 2 + x1 + x2 |
[0] | = | 0 |
[len(x1)] | = | 0 |
[a__len(x1)] | = | -2 |
[s(x1)] | = | -2 |
[nil] | = | 0 |
[cons(x1, x2)] | = | 2 + x1 |
[mark#(x1)] | = | x1 |
mark#(from(X)) | → | a__from#(mark(X)) | (26) |
mark#(cons(X1,X2)) | → | mark#(X1) | (33) |
[mark#(x1)] | = | 1 · x1 |
[a__from#(x1)] | = | 1 · x1 |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
a__from#(X) | → | mark#(X) | (21) |
1 | ≥ | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.