The rewrite relation of the following TRS is considered.
| a__eq(0,0) | → | true | (1) |
| a__eq(s(X),s(Y)) | → | a__eq(X,Y) | (2) |
| a__eq(X,Y) | → | false | (3) |
| a__inf(X) | → | cons(X,inf(s(X))) | (4) |
| a__take(0,X) | → | nil | (5) |
| a__take(s(X),cons(Y,L)) | → | cons(Y,take(X,L)) | (6) |
| a__length(nil) | → | 0 | (7) |
| a__length(cons(X,L)) | → | s(length(L)) | (8) |
| mark(eq(X1,X2)) | → | a__eq(X1,X2) | (9) |
| mark(inf(X)) | → | a__inf(mark(X)) | (10) |
| mark(take(X1,X2)) | → | a__take(mark(X1),mark(X2)) | (11) |
| mark(length(X)) | → | a__length(mark(X)) | (12) |
| mark(0) | → | 0 | (13) |
| mark(true) | → | true | (14) |
| mark(s(X)) | → | s(X) | (15) |
| mark(false) | → | false | (16) |
| mark(cons(X1,X2)) | → | cons(X1,X2) | (17) |
| mark(nil) | → | nil | (18) |
| a__eq(X1,X2) | → | eq(X1,X2) | (19) |
| a__inf(X) | → | inf(X) | (20) |
| a__take(X1,X2) | → | take(X1,X2) | (21) |
| a__length(X) | → | length(X) | (22) |
| prec(mark) | = | 3 | status(mark) | = | [1] | list-extension(mark) | = | Lex | ||
| prec(eq) | = | 0 | status(eq) | = | [2, 1] | list-extension(eq) | = | Lex | ||
| prec(length) | = | 0 | status(length) | = | [1] | list-extension(length) | = | Lex | ||
| prec(a__length) | = | 1 | status(a__length) | = | [1] | list-extension(a__length) | = | Lex | ||
| prec(take) | = | 0 | status(take) | = | [2, 1] | list-extension(take) | = | Lex | ||
| prec(nil) | = | 0 | status(nil) | = | [] | list-extension(nil) | = | Lex | ||
| prec(a__take) | = | 1 | status(a__take) | = | [2, 1] | list-extension(a__take) | = | Lex | ||
| prec(cons) | = | 0 | status(cons) | = | [2, 1] | list-extension(cons) | = | Lex | ||
| prec(inf) | = | 0 | status(inf) | = | [1] | list-extension(inf) | = | Lex | ||
| prec(a__inf) | = | 2 | status(a__inf) | = | [1] | list-extension(a__inf) | = | Lex | ||
| prec(false) | = | 0 | status(false) | = | [] | list-extension(false) | = | Lex | ||
| prec(s) | = | 0 | status(s) | = | [1] | list-extension(s) | = | Lex | ||
| prec(true) | = | 0 | status(true) | = | [] | list-extension(true) | = | Lex | ||
| prec(a__eq) | = | 1 | status(a__eq) | = | [1, 2] | list-extension(a__eq) | = | Lex | ||
| prec(0) | = | 0 | status(0) | = | [] | list-extension(0) | = | Lex |
| [mark(x1)] | = | max(0, 0 + 1 · x1) |
| [eq(x1, x2)] | = | max(0, 0 + 1 · x1, 0 + 1 · x2) |
| [length(x1)] | = | 5 + 1 · x1 |
| [a__length(x1)] | = | max(2, 5 + 1 · x1) |
| [take(x1, x2)] | = | max(0, 0 + 1 · x1, 0 + 1 · x2) |
| [nil] | = | max(3) |
| [a__take(x1, x2)] | = | max(0, 0 + 1 · x1, 0 + 1 · x2) |
| [cons(x1, x2)] | = | max(0, 0 + 1 · x1, 0 + 1 · x2) |
| [inf(x1)] | = | max(0, 0 + 1 · x1) |
| [a__inf(x1)] | = | max(0, 0 + 1 · x1) |
| [false] | = | max(0) |
| [s(x1)] | = | max(0, 0 + 1 · x1) |
| [true] | = | max(0) |
| [a__eq(x1, x2)] | = | max(0, 0 + 1 · x1, 0 + 1 · x2) |
| [0] | = | max(4) |
| a__eq(0,0) | → | true | (1) |
| a__eq(s(X),s(Y)) | → | a__eq(X,Y) | (2) |
| a__eq(X,Y) | → | false | (3) |
| a__inf(X) | → | cons(X,inf(s(X))) | (4) |
| a__take(0,X) | → | nil | (5) |
| a__take(s(X),cons(Y,L)) | → | cons(Y,take(X,L)) | (6) |
| a__length(nil) | → | 0 | (7) |
| a__length(cons(X,L)) | → | s(length(L)) | (8) |
| mark(eq(X1,X2)) | → | a__eq(X1,X2) | (9) |
| mark(inf(X)) | → | a__inf(mark(X)) | (10) |
| mark(take(X1,X2)) | → | a__take(mark(X1),mark(X2)) | (11) |
| mark(length(X)) | → | a__length(mark(X)) | (12) |
| mark(0) | → | 0 | (13) |
| mark(true) | → | true | (14) |
| mark(s(X)) | → | s(X) | (15) |
| mark(false) | → | false | (16) |
| mark(cons(X1,X2)) | → | cons(X1,X2) | (17) |
| mark(nil) | → | nil | (18) |
| a__eq(X1,X2) | → | eq(X1,X2) | (19) |
| a__inf(X) | → | inf(X) | (20) |
| a__take(X1,X2) | → | take(X1,X2) | (21) |
| a__length(X) | → | length(X) | (22) |
There are no rules in the TRS. Hence, it is terminating.