The rewrite relation of the following TRS is considered.
| from(X) | → | cons(X,n__from(n__s(X))) | (1) |
| 2ndspos(0,Z) | → | rnil | (2) |
| 2ndspos(s(N),cons(X,n__cons(Y,Z))) | → | rcons(posrecip(activate(Y)),2ndsneg(N,activate(Z))) | (3) |
| 2ndsneg(0,Z) | → | rnil | (4) |
| 2ndsneg(s(N),cons(X,n__cons(Y,Z))) | → | rcons(negrecip(activate(Y)),2ndspos(N,activate(Z))) | (5) |
| pi(X) | → | 2ndspos(X,from(0)) | (6) |
| plus(0,Y) | → | Y | (7) |
| plus(s(X),Y) | → | s(plus(X,Y)) | (8) |
| times(0,Y) | → | 0 | (9) |
| times(s(X),Y) | → | plus(Y,times(X,Y)) | (10) |
| square(X) | → | times(X,X) | (11) |
| from(X) | → | n__from(X) | (12) |
| s(X) | → | n__s(X) | (13) |
| cons(X1,X2) | → | n__cons(X1,X2) | (14) |
| activate(n__from(X)) | → | from(activate(X)) | (15) |
| activate(n__s(X)) | → | s(activate(X)) | (16) |
| activate(n__cons(X1,X2)) | → | cons(activate(X1),X2) | (17) |
| activate(X) | → | X | (18) |
| from#(X) | → | cons#(X,n__from(n__s(X))) | (19) |
| 2ndspos#(s(N),cons(X,n__cons(Y,Z))) | → | activate#(Z) | (20) |
| 2ndspos#(s(N),cons(X,n__cons(Y,Z))) | → | 2ndsneg#(N,activate(Z)) | (21) |
| 2ndspos#(s(N),cons(X,n__cons(Y,Z))) | → | activate#(Y) | (22) |
| 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) | → | activate#(Z) | (23) |
| 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) | → | 2ndspos#(N,activate(Z)) | (24) |
| 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) | → | activate#(Y) | (25) |
| pi#(X) | → | from#(0) | (26) |
| pi#(X) | → | 2ndspos#(X,from(0)) | (27) |
| plus#(s(X),Y) | → | plus#(X,Y) | (28) |
| plus#(s(X),Y) | → | s#(plus(X,Y)) | (29) |
| times#(s(X),Y) | → | times#(X,Y) | (30) |
| times#(s(X),Y) | → | plus#(Y,times(X,Y)) | (31) |
| square#(X) | → | times#(X,X) | (32) |
| activate#(n__from(X)) | → | activate#(X) | (33) |
| activate#(n__from(X)) | → | from#(activate(X)) | (34) |
| activate#(n__s(X)) | → | activate#(X) | (35) |
| activate#(n__s(X)) | → | s#(activate(X)) | (36) |
| activate#(n__cons(X1,X2)) | → | activate#(X1) | (37) |
| activate#(n__cons(X1,X2)) | → | cons#(activate(X1),X2) | (38) |
The dependency pairs are split into 4 components.
| 2ndspos#(s(N),cons(X,n__cons(Y,Z))) | → | 2ndsneg#(N,activate(Z)) | (21) |
| 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) | → | 2ndspos#(N,activate(Z)) | (24) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
| 2ndspos#(s(N),cons(X,n__cons(Y,Z))) | → | 2ndsneg#(N,activate(Z)) | (21) |
| 1 | > | 1 | |
| 2ndsneg#(s(N),cons(X,n__cons(Y,Z))) | → | 2ndspos#(N,activate(Z)) | (24) |
| 1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
| activate#(n__from(X)) | → | activate#(X) | (33) |
| activate#(n__s(X)) | → | activate#(X) | (35) |
| activate#(n__cons(X1,X2)) | → | activate#(X1) | (37) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
| activate#(n__from(X)) | → | activate#(X) | (33) |
| 1 | > | 1 | |
| activate#(n__s(X)) | → | activate#(X) | (35) |
| 1 | > | 1 | |
| activate#(n__cons(X1,X2)) | → | activate#(X1) | (37) |
| 1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
| times#(s(X),Y) | → | times#(X,Y) | (30) |
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) | (30) |
| 2 | ≥ | 2 | |
| 1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
| plus#(s(X),Y) | → | plus#(X,Y) | (28) |
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) | (28) |
| 2 | ≥ | 2 | |
| 1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.