The rewrite relation of the following TRS is considered.
| a__terms(N) | → | cons(recip(a__sqr(mark(N))),terms(s(N))) | (1) |
| a__sqr(0) | → | 0 | (2) |
| a__sqr(s(X)) | → | s(add(sqr(X),dbl(X))) | (3) |
| a__dbl(0) | → | 0 | (4) |
| a__dbl(s(X)) | → | s(s(dbl(X))) | (5) |
| a__add(0,X) | → | mark(X) | (6) |
| a__add(s(X),Y) | → | s(add(X,Y)) | (7) |
| a__first(0,X) | → | nil | (8) |
| a__first(s(X),cons(Y,Z)) | → | cons(mark(Y),first(X,Z)) | (9) |
| mark(terms(X)) | → | a__terms(mark(X)) | (10) |
| mark(sqr(X)) | → | a__sqr(mark(X)) | (11) |
| mark(add(X1,X2)) | → | a__add(mark(X1),mark(X2)) | (12) |
| mark(dbl(X)) | → | a__dbl(mark(X)) | (13) |
| mark(first(X1,X2)) | → | a__first(mark(X1),mark(X2)) | (14) |
| mark(cons(X1,X2)) | → | cons(mark(X1),X2) | (15) |
| mark(recip(X)) | → | recip(mark(X)) | (16) |
| mark(s(X)) | → | s(X) | (17) |
| mark(0) | → | 0 | (18) |
| mark(nil) | → | nil | (19) |
| a__terms(X) | → | terms(X) | (20) |
| a__sqr(X) | → | sqr(X) | (21) |
| a__add(X1,X2) | → | add(X1,X2) | (22) |
| a__dbl(X) | → | dbl(X) | (23) |
| a__first(X1,X2) | → | first(X1,X2) | (24) |
| a__terms#(N) | → | a__sqr#(mark(N)) | (25) |
| a__terms#(N) | → | mark#(N) | (26) |
| a__add#(0,X) | → | mark#(X) | (27) |
| a__first#(s(X),cons(Y,Z)) | → | mark#(Y) | (28) |
| mark#(terms(X)) | → | a__terms#(mark(X)) | (29) |
| mark#(terms(X)) | → | mark#(X) | (30) |
| mark#(sqr(X)) | → | a__sqr#(mark(X)) | (31) |
| mark#(sqr(X)) | → | mark#(X) | (32) |
| mark#(add(X1,X2)) | → | a__add#(mark(X1),mark(X2)) | (33) |
| mark#(add(X1,X2)) | → | mark#(X1) | (34) |
| mark#(add(X1,X2)) | → | mark#(X2) | (35) |
| mark#(dbl(X)) | → | a__dbl#(mark(X)) | (36) |
| mark#(dbl(X)) | → | mark#(X) | (37) |
| mark#(first(X1,X2)) | → | a__first#(mark(X1),mark(X2)) | (38) |
| mark#(first(X1,X2)) | → | mark#(X1) | (39) |
| mark#(first(X1,X2)) | → | mark#(X2) | (40) |
| mark#(cons(X1,X2)) | → | mark#(X1) | (41) |
| mark#(recip(X)) | → | mark#(X) | (42) |
The dependency pairs are split into 1 component.
| a__terms#(N) | → | mark#(N) | (26) |
| mark#(terms(X)) | → | a__terms#(mark(X)) | (29) |
| mark#(terms(X)) | → | mark#(X) | (30) |
| mark#(sqr(X)) | → | mark#(X) | (32) |
| mark#(add(X1,X2)) | → | a__add#(mark(X1),mark(X2)) | (33) |
| a__add#(0,X) | → | mark#(X) | (27) |
| mark#(add(X1,X2)) | → | mark#(X1) | (34) |
| mark#(add(X1,X2)) | → | mark#(X2) | (35) |
| mark#(dbl(X)) | → | mark#(X) | (37) |
| mark#(first(X1,X2)) | → | a__first#(mark(X1),mark(X2)) | (38) |
| a__first#(s(X),cons(Y,Z)) | → | mark#(Y) | (28) |
| mark#(first(X1,X2)) | → | mark#(X1) | (39) |
| mark#(first(X1,X2)) | → | mark#(X2) | (40) |
| mark#(cons(X1,X2)) | → | mark#(X1) | (41) |
| mark#(recip(X)) | → | mark#(X) | (42) |
| [a__add#(x1, x2)] | = | 1 + 2 · x2 |
| [a__first#(x1, x2)] | = | 2 + x1 + 2 · x2 |
| [a__terms#(x1)] | = | 1 + 2 · x1 |
| [mark(x1)] | = | x1 |
| [terms(x1)] | = | 1 + 2 · x1 |
| [a__terms(x1)] | = | 1 + 2 · x1 |
| [sqr(x1)] | = | x1 |
| [a__sqr(x1)] | = | x1 |
| [add(x1, x2)] | = | 1 + 2 · x1 + 2 · x2 |
| [a__add(x1, x2)] | = | 1 + 2 · x1 + 2 · x2 |
| [0] | = | 0 |
| [dbl(x1)] | = | x1 |
| [a__dbl(x1)] | = | x1 |
| [first(x1, x2)] | = | 1 + x1 + x2 |
| [a__first(x1, x2)] | = | 1 + x1 + x2 |
| [cons(x1, x2)] | = | 1 + x1 |
| [recip(x1)] | = | x1 |
| [s(x1)] | = | 2 |
| [nil] | = | 0 |
| [mark#(x1)] | = | 2 · x1 |
| a__terms#(N) | → | mark#(N) | (26) |
| mark#(terms(X)) | → | a__terms#(mark(X)) | (29) |
| mark#(terms(X)) | → | mark#(X) | (30) |
| mark#(add(X1,X2)) | → | a__add#(mark(X1),mark(X2)) | (33) |
| a__add#(0,X) | → | mark#(X) | (27) |
| mark#(add(X1,X2)) | → | mark#(X1) | (34) |
| mark#(add(X1,X2)) | → | mark#(X2) | (35) |
| a__first#(s(X),cons(Y,Z)) | → | mark#(Y) | (28) |
| mark#(first(X1,X2)) | → | mark#(X1) | (39) |
| mark#(first(X1,X2)) | → | mark#(X2) | (40) |
| mark#(cons(X1,X2)) | → | mark#(X1) | (41) |
The dependency pairs are split into 1 component.
| mark#(dbl(X)) | → | mark#(X) | (37) |
| mark#(sqr(X)) | → | mark#(X) | (32) |
| mark#(recip(X)) | → | mark#(X) | (42) |
| [dbl(x1)] | = | 1 · x1 |
| [sqr(x1)] | = | 1 · x1 |
| [recip(x1)] | = | 1 · x1 |
| [mark#(x1)] | = | 1 · x1 |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
| mark#(dbl(X)) | → | mark#(X) | (37) |
| 1 | > | 1 | |
| mark#(sqr(X)) | → | mark#(X) | (32) |
| 1 | > | 1 | |
| mark#(recip(X)) | → | mark#(X) | (42) |
| 1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.