The rewrite relation of the following TRS is considered.
from(X) | → | cons(X,n__from(s(X))) | (1) |
2ndspos(0,Z) | → | rnil | (2) |
2ndspos(s(N),cons(X,Z)) | → | 2ndspos(s(N),cons2(X,activate(Z))) | (3) |
2ndspos(s(N),cons2(X,cons(Y,Z))) | → | rcons(posrecip(Y),2ndsneg(N,activate(Z))) | (4) |
2ndsneg(0,Z) | → | rnil | (5) |
2ndsneg(s(N),cons(X,Z)) | → | 2ndsneg(s(N),cons2(X,activate(Z))) | (6) |
2ndsneg(s(N),cons2(X,cons(Y,Z))) | → | rcons(negrecip(Y),2ndspos(N,activate(Z))) | (7) |
pi(X) | → | 2ndspos(X,from(0)) | (8) |
plus(0,Y) | → | Y | (9) |
plus(s(X),Y) | → | s(plus(X,Y)) | (10) |
times(0,Y) | → | 0 | (11) |
times(s(X),Y) | → | plus(Y,times(X,Y)) | (12) |
square(X) | → | times(X,X) | (13) |
from(X) | → | n__from(X) | (14) |
activate(n__from(X)) | → | from(X) | (15) |
activate(X) | → | X | (16) |
2ndspos#(s(N),cons(X,Z)) | → | 2ndspos#(s(N),cons2(X,activate(Z))) | (17) |
2ndspos#(s(N),cons(X,Z)) | → | activate#(Z) | (18) |
2ndspos#(s(N),cons2(X,cons(Y,Z))) | → | 2ndsneg#(N,activate(Z)) | (19) |
2ndspos#(s(N),cons2(X,cons(Y,Z))) | → | activate#(Z) | (20) |
2ndsneg#(s(N),cons(X,Z)) | → | 2ndsneg#(s(N),cons2(X,activate(Z))) | (21) |
2ndsneg#(s(N),cons(X,Z)) | → | activate#(Z) | (22) |
2ndsneg#(s(N),cons2(X,cons(Y,Z))) | → | 2ndspos#(N,activate(Z)) | (23) |
2ndsneg#(s(N),cons2(X,cons(Y,Z))) | → | activate#(Z) | (24) |
pi#(X) | → | 2ndspos#(X,from(0)) | (25) |
pi#(X) | → | from#(0) | (26) |
plus#(s(X),Y) | → | plus#(X,Y) | (27) |
times#(s(X),Y) | → | plus#(Y,times(X,Y)) | (28) |
times#(s(X),Y) | → | times#(X,Y) | (29) |
square#(X) | → | times#(X,X) | (30) |
activate#(n__from(X)) | → | from#(X) | (31) |
The dependency pairs are split into 3 components.
times#(s(X),Y) | → | times#(X,Y) | (29) |
[s(x1)] | = | 1 · x1 |
[times#(x1, x2)] | = | 1 · x1 + 1 · x2 |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
times#(s(X),Y) | → | times#(X,Y) | (29) |
1 | > | 1 | |
2 | ≥ | 2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
2ndspos#(s(N),cons2(X,cons(Y,Z))) | → | 2ndsneg#(N,activate(Z)) | (19) |
2ndsneg#(s(N),cons(X,Z)) | → | 2ndsneg#(s(N),cons2(X,activate(Z))) | (21) |
2ndsneg#(s(N),cons2(X,cons(Y,Z))) | → | 2ndspos#(N,activate(Z)) | (23) |
2ndspos#(s(N),cons(X,Z)) | → | 2ndspos#(s(N),cons2(X,activate(Z))) | (17) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
2ndsneg#(s(N),cons2(X,cons(Y,Z))) | → | 2ndspos#(N,activate(Z)) | (23) |
1 | > | 1 | |
2ndsneg#(s(N),cons(X,Z)) | → | 2ndsneg#(s(N),cons2(X,activate(Z))) | (21) |
1 | ≥ | 1 | |
2ndspos#(s(N),cons(X,Z)) | → | 2ndspos#(s(N),cons2(X,activate(Z))) | (17) |
1 | ≥ | 1 | |
2ndspos#(s(N),cons2(X,cons(Y,Z))) | → | 2ndsneg#(N,activate(Z)) | (19) |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
plus#(s(X),Y) | → | plus#(X,Y) | (27) |
[s(x1)] | = | 1 · x1 |
[plus#(x1, x2)] | = | 1 · x1 + 1 · x2 |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
plus#(s(X),Y) | → | plus#(X,Y) | (27) |
1 | > | 1 | |
2 | ≥ | 2 |
As there is no critical graph in the transitive closure, there are no infinite chains.