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.