The rewrite relation of the following TRS is considered.
| first(0,X) | → | nil | (1) |
| first(s(X),cons(Y,Z)) | → | cons(Y,n__first(X,activate(Z))) | (2) |
| from(X) | → | cons(X,n__from(n__s(X))) | (3) |
| first(X1,X2) | → | n__first(X1,X2) | (4) |
| from(X) | → | n__from(X) | (5) |
| s(X) | → | n__s(X) | (6) |
| activate(n__first(X1,X2)) | → | first(activate(X1),activate(X2)) | (7) |
| activate(n__from(X)) | → | from(activate(X)) | (8) |
| activate(n__s(X)) | → | s(activate(X)) | (9) |
| activate(X) | → | X | (10) |
| first#(s(X),cons(Y,Z)) | → | activate#(Z) | (11) |
| activate#(n__first(X1,X2)) | → | first#(activate(X1),activate(X2)) | (12) |
| activate#(n__first(X1,X2)) | → | activate#(X1) | (13) |
| activate#(n__first(X1,X2)) | → | activate#(X2) | (14) |
| activate#(n__from(X)) | → | from#(activate(X)) | (15) |
| activate#(n__from(X)) | → | activate#(X) | (16) |
| activate#(n__s(X)) | → | s#(activate(X)) | (17) |
| activate#(n__s(X)) | → | activate#(X) | (18) |
The dependency pairs are split into 1 component.
| activate#(n__first(X1,X2)) | → | first#(activate(X1),activate(X2)) | (12) |
| first#(s(X),cons(Y,Z)) | → | activate#(Z) | (11) |
| activate#(n__first(X1,X2)) | → | activate#(X1) | (13) |
| activate#(n__first(X1,X2)) | → | activate#(X2) | (14) |
| activate#(n__from(X)) | → | activate#(X) | (16) |
| activate#(n__s(X)) | → | activate#(X) | (18) |
| [first#(x1, x2)] | = | 1 + x1 + x2 |
| [activate(x1)] | = | x1 |
| [n__first(x1, x2)] | = | 1 + x1 + x2 |
| [first(x1, x2)] | = | 1 + x1 + x2 |
| [n__from(x1)] | = | 2 + x1 |
| [from(x1)] | = | 2 + x1 |
| [n__s(x1)] | = | 1 + x1 |
| [s(x1)] | = | 1 + x1 |
| [cons(x1, x2)] | = | -2 + x2 |
| [0] | = | 0 |
| [nil] | = | 1 |
| [activate#(x1)] | = | x1 |
| activate#(n__first(X1,X2)) | → | activate#(X1) | (13) |
| activate#(n__first(X1,X2)) | → | activate#(X2) | (14) |
| activate#(n__from(X)) | → | activate#(X) | (16) |
| activate#(n__s(X)) | → | activate#(X) | (18) |
| prec(n__first) | = | 2 | weight(n__first) | = | 2 | ||||
| prec(activate) | = | 7 | weight(activate) | = | 0 | ||||
| prec(first) | = | 4 | weight(first) | = | 2 | ||||
| prec(n__from) | = | 3 | weight(n__from) | = | 2 | ||||
| prec(from) | = | 5 | weight(from) | = | 2 | ||||
| prec(n__s) | = | 0 | weight(n__s) | = | 1 | ||||
| prec(s) | = | 1 | weight(s) | = | 1 | ||||
| prec(nil) | = | 6 | weight(nil) | = | 2 |
| π(activate#) | = | 1 |
| π(n__first) | = | [2] |
| π(first#) | = | 2 |
| π(activate) | = | [1] |
| π(cons) | = | 2 |
| π(first) | = | [2] |
| π(n__from) | = | [] |
| π(from) | = | [] |
| π(n__s) | = | [] |
| π(s) | = | [] |
| π(nil) | = | [] |
| activate#(n__first(X1,X2)) | → | first#(activate(X1),activate(X2)) | (12) |
| [s(x1)] | = | 1 · x1 |
| [cons(x1, x2)] | = | 1 · x1 + 1 · x2 |
| [activate#(x1)] | = | 1 · x1 |
| [first#(x1, x2)] | = | 1 · x1 + 1 · x2 |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
| first#(s(X),cons(Y,Z)) | → | activate#(Z) | (11) |
| 2 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.