The rewrite relation of the following TRS is considered.
from(X) | → | cons(X,n__from(n__s(X))) | (1) |
first(0,Z) | → | nil | (2) |
first(s(X),cons(Y,Z)) | → | cons(Y,n__first(X,activate(Z))) | (3) |
sel(0,cons(X,Z)) | → | X | (4) |
sel(s(X),cons(Y,Z)) | → | sel(X,activate(Z)) | (5) |
from(X) | → | n__from(X) | (6) |
s(X) | → | n__s(X) | (7) |
first(X1,X2) | → | n__first(X1,X2) | (8) |
activate(n__from(X)) | → | from(activate(X)) | (9) |
activate(n__s(X)) | → | s(activate(X)) | (10) |
activate(n__first(X1,X2)) | → | first(activate(X1),activate(X2)) | (11) |
activate(X) | → | X | (12) |
first#(s(X),cons(Y,Z)) | → | activate#(Z) | (13) |
sel#(s(X),cons(Y,Z)) | → | sel#(X,activate(Z)) | (14) |
sel#(s(X),cons(Y,Z)) | → | activate#(Z) | (15) |
activate#(n__from(X)) | → | from#(activate(X)) | (16) |
activate#(n__from(X)) | → | activate#(X) | (17) |
activate#(n__s(X)) | → | s#(activate(X)) | (18) |
activate#(n__s(X)) | → | activate#(X) | (19) |
activate#(n__first(X1,X2)) | → | first#(activate(X1),activate(X2)) | (20) |
activate#(n__first(X1,X2)) | → | activate#(X1) | (21) |
activate#(n__first(X1,X2)) | → | activate#(X2) | (22) |
The dependency pairs are split into 2 components.
sel#(s(X),cons(Y,Z)) | → | sel#(X,activate(Z)) | (14) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
sel#(s(X),cons(Y,Z)) | → | sel#(X,activate(Z)) | (14) |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
activate#(n__from(X)) | → | activate#(X) | (17) |
activate#(n__s(X)) | → | activate#(X) | (19) |
activate#(n__first(X1,X2)) | → | first#(activate(X1),activate(X2)) | (20) |
first#(s(X),cons(Y,Z)) | → | activate#(Z) | (13) |
activate#(n__first(X1,X2)) | → | activate#(X1) | (21) |
activate#(n__first(X1,X2)) | → | activate#(X2) | (22) |
prec(n__first) | = | 0 | weight(n__first) | = | 2 | ||||
prec(activate) | = | 4 | weight(activate) | = | 0 | ||||
prec(first) | = | 2 | weight(first) | = | 2 | ||||
prec(0) | = | 1 | weight(0) | = | 2 | ||||
prec(nil) | = | 3 | weight(nil) | = | 4 |
π(activate#) | = | 1 |
π(n__from) | = | 1 |
π(n__s) | = | 1 |
π(n__first) | = | [1,2] |
π(first#) | = | 2 |
π(activate) | = | [1] |
π(cons) | = | 2 |
π(from) | = | 1 |
π(s) | = | 1 |
π(first) | = | [1,2] |
π(0) | = | [] |
π(nil) | = | [] |
activate(n__from(X)) | → | from(activate(X)) | (9) |
activate(n__s(X)) | → | s(activate(X)) | (10) |
activate(n__first(X1,X2)) | → | first(activate(X1),activate(X2)) | (11) |
activate(X) | → | X | (12) |
first(s(X),cons(Y,Z)) | → | cons(Y,n__first(X,activate(Z))) | (3) |
s(X) | → | n__s(X) | (7) |
from(X) | → | cons(X,n__from(n__s(X))) | (1) |
from(X) | → | n__from(X) | (6) |
first(0,Z) | → | nil | (2) |
first(X1,X2) | → | n__first(X1,X2) | (8) |
activate#(n__first(X1,X2)) | → | first#(activate(X1),activate(X2)) | (20) |
activate#(n__first(X1,X2)) | → | activate#(X1) | (21) |
activate#(n__first(X1,X2)) | → | activate#(X2) | (22) |
The dependency pairs are split into 1 component.
activate#(n__s(X)) | → | activate#(X) | (19) |
activate#(n__from(X)) | → | activate#(X) | (17) |
[n__s(x1)] | = | 1 · x1 |
[n__from(x1)] | = | 1 · x1 |
[activate#(x1)] | = | 1 · x1 |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
activate#(n__s(X)) | → | activate#(X) | (19) |
1 | > | 1 | |
activate#(n__from(X)) | → | activate#(X) | (17) |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.