The rewrite relation of the following TRS is considered.
a__first(0,X) | → | nil | (1) |
a__first(s(X),cons(Y,Z)) | → | cons(mark(Y),first(X,Z)) | (2) |
a__from(X) | → | cons(mark(X),from(s(X))) | (3) |
mark(first(X1,X2)) | → | a__first(mark(X1),mark(X2)) | (4) |
mark(from(X)) | → | a__from(mark(X)) | (5) |
mark(0) | → | 0 | (6) |
mark(nil) | → | nil | (7) |
mark(s(X)) | → | s(mark(X)) | (8) |
mark(cons(X1,X2)) | → | cons(mark(X1),X2) | (9) |
a__first(X1,X2) | → | first(X1,X2) | (10) |
a__from(X) | → | from(X) | (11) |
prec(from) | = | 0 | status(from) | = | [1] | list-extension(from) | = | Lex | ||
prec(a__from) | = | 1 | status(a__from) | = | [1] | list-extension(a__from) | = | Lex | ||
prec(first) | = | 0 | status(first) | = | [2, 1] | list-extension(first) | = | Lex | ||
prec(mark) | = | 8 | status(mark) | = | [1] | list-extension(mark) | = | Lex | ||
prec(cons) | = | 0 | status(cons) | = | [1, 2] | list-extension(cons) | = | Lex | ||
prec(s) | = | 0 | status(s) | = | [1] | list-extension(s) | = | Lex | ||
prec(nil) | = | 0 | status(nil) | = | [] | list-extension(nil) | = | Lex | ||
prec(a__first) | = | 1 | status(a__first) | = | [1, 2] | list-extension(a__first) | = | Lex | ||
prec(0) | = | 0 | status(0) | = | [] | list-extension(0) | = | Lex |
[from(x1)] | = | max(0, 1 + 1 · x1) |
[a__from(x1)] | = | max(1, 1 + 1 · x1) |
[first(x1, x2)] | = | max(0, 0 + 1 · x1, 7 + 1 · x2) |
[mark(x1)] | = | 0 + 1 · x1 |
[cons(x1, x2)] | = | max(1, 0 + 1 · x1, 0 + 1 · x2) |
[s(x1)] | = | max(0, 0 + 1 · x1) |
[nil] | = | max(4) |
[a__first(x1, x2)] | = | max(0, 0 + 1 · x1, 7 + 1 · x2) |
[0] | = | max(0) |
a__first(0,X) | → | nil | (1) |
a__first(s(X),cons(Y,Z)) | → | cons(mark(Y),first(X,Z)) | (2) |
a__from(X) | → | cons(mark(X),from(s(X))) | (3) |
mark(first(X1,X2)) | → | a__first(mark(X1),mark(X2)) | (4) |
mark(from(X)) | → | a__from(mark(X)) | (5) |
mark(0) | → | 0 | (6) |
mark(nil) | → | nil | (7) |
mark(s(X)) | → | s(mark(X)) | (8) |
mark(cons(X1,X2)) | → | cons(mark(X1),X2) | (9) |
a__first(X1,X2) | → | first(X1,X2) | (10) |
a__from(X) | → | from(X) | (11) |
There are no rules in the TRS. Hence, it is terminating.