The rewrite relation of the following TRS is considered.
| and(true,X) | → | activate(X) | (1) |
| and(false,Y) | → | false | (2) |
| if(true,X,Y) | → | activate(X) | (3) |
| if(false,X,Y) | → | activate(Y) | (4) |
| add(0,X) | → | activate(X) | (5) |
| add(s(X),Y) | → | s(n__add(activate(X),activate(Y))) | (6) |
| first(0,X) | → | nil | (7) |
| first(s(X),cons(Y,Z)) | → | cons(activate(Y),n__first(activate(X),activate(Z))) | (8) |
| from(X) | → | cons(activate(X),n__from(n__s(activate(X)))) | (9) |
| add(X1,X2) | → | n__add(X1,X2) | (10) |
| first(X1,X2) | → | n__first(X1,X2) | (11) |
| from(X) | → | n__from(X) | (12) |
| s(X) | → | n__s(X) | (13) |
| activate(n__add(X1,X2)) | → | add(activate(X1),X2) | (14) |
| activate(n__first(X1,X2)) | → | first(activate(X1),activate(X2)) | (15) |
| activate(n__from(X)) | → | from(X) | (16) |
| activate(n__s(X)) | → | s(X) | (17) |
| activate(X) | → | X | (18) |
| prec(n__from) | = | 0 | status(n__from) | = | [1] | list-extension(n__from) | = | Lex | ||
| prec(n__s) | = | 0 | status(n__s) | = | [1] | list-extension(n__s) | = | Lex | ||
| prec(from) | = | 1 | status(from) | = | [1] | list-extension(from) | = | Lex | ||
| prec(n__first) | = | 0 | status(n__first) | = | [1, 2] | list-extension(n__first) | = | Lex | ||
| prec(cons) | = | 0 | status(cons) | = | [1, 2] | list-extension(cons) | = | Lex | ||
| prec(nil) | = | 0 | status(nil) | = | [] | list-extension(nil) | = | Lex | ||
| prec(first) | = | 1 | status(first) | = | [2, 1] | list-extension(first) | = | Lex | ||
| prec(n__add) | = | 0 | status(n__add) | = | [1, 2] | list-extension(n__add) | = | Lex | ||
| prec(s) | = | 2 | status(s) | = | [1] | list-extension(s) | = | Lex | ||
| prec(add) | = | 3 | status(add) | = | [2, 1] | list-extension(add) | = | Lex | ||
| prec(0) | = | 0 | status(0) | = | [] | list-extension(0) | = | Lex | ||
| prec(if) | = | 0 | status(if) | = | [3, 2, 1] | list-extension(if) | = | Lex | ||
| prec(false) | = | 0 | status(false) | = | [] | list-extension(false) | = | Lex | ||
| prec(activate) | = | 4 | status(activate) | = | [1] | list-extension(activate) | = | Lex | ||
| prec(and) | = | 0 | status(and) | = | [1, 2] | list-extension(and) | = | Lex | ||
| prec(true) | = | 0 | status(true) | = | [] | list-extension(true) | = | Lex |
| [n__from(x1)] | = | max(1, 1 + 1 · x1) |
| [n__s(x1)] | = | 0 + 1 · x1 |
| [from(x1)] | = | max(0, 1 + 1 · x1) |
| [n__first(x1, x2)] | = | max(1, 3 + 1 · x1, 2 + 1 · x2) |
| [cons(x1, x2)] | = | max(0, 1 + 1 · x1, 0 + 1 · x2) |
| [nil] | = | max(1) |
| [first(x1, x2)] | = | max(3, 3 + 1 · x1, 2 + 1 · x2) |
| [n__add(x1, x2)] | = | max(0, 2 + 1 · x1, 1 + 1 · x2) |
| [s(x1)] | = | max(0, 0 + 1 · x1) |
| [add(x1, x2)] | = | max(0, 2 + 1 · x1, 1 + 1 · x2) |
| [0] | = | max(0) |
| [if(x1, x2, x3)] | = | max(6, 5 + 1 · x1, 5 + 1 · x2, 5 + 1 · x3) |
| [false] | = | max(0) |
| [activate(x1)] | = | max(0, 0 + 1 · x1) |
| [and(x1, x2)] | = | 1 + 1 · x1 + 1 · x2 |
| [true] | = | max(0) |
| and(true,X) | → | activate(X) | (1) |
| and(false,Y) | → | false | (2) |
| if(true,X,Y) | → | activate(X) | (3) |
| if(false,X,Y) | → | activate(Y) | (4) |
| add(0,X) | → | activate(X) | (5) |
| add(s(X),Y) | → | s(n__add(activate(X),activate(Y))) | (6) |
| first(0,X) | → | nil | (7) |
| first(s(X),cons(Y,Z)) | → | cons(activate(Y),n__first(activate(X),activate(Z))) | (8) |
| from(X) | → | cons(activate(X),n__from(n__s(activate(X)))) | (9) |
| add(X1,X2) | → | n__add(X1,X2) | (10) |
| first(X1,X2) | → | n__first(X1,X2) | (11) |
| from(X) | → | n__from(X) | (12) |
| s(X) | → | n__s(X) | (13) |
| activate(n__add(X1,X2)) | → | add(activate(X1),X2) | (14) |
| activate(n__first(X1,X2)) | → | first(activate(X1),activate(X2)) | (15) |
| activate(n__from(X)) | → | from(X) | (16) |
| activate(n__s(X)) | → | s(X) | (17) |
| activate(X) | → | X | (18) |
There are no rules in the TRS. Hence, it is terminating.