The rewrite relation of the following TRS is considered.
from(X) | → | cons(X,n__from(s(X))) | (1) |
sel(0,cons(X,XS)) | → | X | (2) |
sel(s(N),cons(X,XS)) | → | sel(N,activate(XS)) | (3) |
minus(X,0) | → | 0 | (4) |
minus(s(X),s(Y)) | → | minus(X,Y) | (5) |
quot(0,s(Y)) | → | 0 | (6) |
quot(s(X),s(Y)) | → | s(quot(minus(X,Y),s(Y))) | (7) |
zWquot(XS,nil) | → | nil | (8) |
zWquot(nil,XS) | → | nil | (9) |
zWquot(cons(X,XS),cons(Y,YS)) | → | cons(quot(X,Y),n__zWquot(activate(XS),activate(YS))) | (10) |
from(X) | → | n__from(X) | (11) |
zWquot(X1,X2) | → | n__zWquot(X1,X2) | (12) |
activate(n__from(X)) | → | from(X) | (13) |
activate(n__zWquot(X1,X2)) | → | zWquot(X1,X2) | (14) |
activate(X) | → | X | (15) |
sel#(s(N),cons(X,XS)) | → | sel#(N,activate(XS)) | (16) |
sel#(s(N),cons(X,XS)) | → | activate#(XS) | (17) |
minus#(s(X),s(Y)) | → | minus#(X,Y) | (18) |
quot#(s(X),s(Y)) | → | quot#(minus(X,Y),s(Y)) | (19) |
quot#(s(X),s(Y)) | → | minus#(X,Y) | (20) |
zWquot#(cons(X,XS),cons(Y,YS)) | → | quot#(X,Y) | (21) |
zWquot#(cons(X,XS),cons(Y,YS)) | → | activate#(XS) | (22) |
zWquot#(cons(X,XS),cons(Y,YS)) | → | activate#(YS) | (23) |
activate#(n__from(X)) | → | from#(X) | (24) |
activate#(n__zWquot(X1,X2)) | → | zWquot#(X1,X2) | (25) |
The dependency pairs are split into 4 components.
sel#(s(N),cons(X,XS)) | → | sel#(N,activate(XS)) | (16) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
sel#(s(N),cons(X,XS)) | → | sel#(N,activate(XS)) | (16) |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
zWquot#(cons(X,XS),cons(Y,YS)) | → | activate#(XS) | (22) |
activate#(n__zWquot(X1,X2)) | → | zWquot#(X1,X2) | (25) |
zWquot#(cons(X,XS),cons(Y,YS)) | → | activate#(YS) | (23) |
[cons(x1, x2)] | = | 1 · x1 + 1 · x2 |
[n__zWquot(x1, x2)] | = | 1 · x1 + 1 · x2 |
[activate#(x1)] | = | 1 · x1 |
[zWquot#(x1, x2)] | = | 1 · x1 + 1 · x2 |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
activate#(n__zWquot(X1,X2)) | → | zWquot#(X1,X2) | (25) |
1 | > | 1 | |
1 | > | 2 | |
zWquot#(cons(X,XS),cons(Y,YS)) | → | activate#(XS) | (22) |
1 | > | 1 | |
zWquot#(cons(X,XS),cons(Y,YS)) | → | activate#(YS) | (23) |
2 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
quot#(s(X),s(Y)) | → | quot#(minus(X,Y),s(Y)) | (19) |
prec(s) | = | 2 | weight(s) | = | 1 | ||||
prec(minus) | = | 1 | weight(minus) | = | 1 | ||||
prec(0) | = | 0 | weight(0) | = | 1 |
π(quot#) | = | 1 |
π(s) | = | [] |
π(minus) | = | [] |
π(0) | = | [] |
minus(X,0) | → | 0 | (4) |
minus(s(X),s(Y)) | → | minus(X,Y) | (5) |
quot#(s(X),s(Y)) | → | quot#(minus(X,Y),s(Y)) | (19) |
There are no pairs anymore.
minus#(s(X),s(Y)) | → | minus#(X,Y) | (18) |
[s(x1)] | = | 1 · x1 |
[minus#(x1, x2)] | = | 1 · x1 + 1 · x2 |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
minus#(s(X),s(Y)) | → | minus#(X,Y) | (18) |
1 | > | 1 | |
2 | > | 2 |
As there is no critical graph in the transitive closure, there are no infinite chains.